Of course. Iteration is the process of repeating a procedure or a set of steps to achieve a goal. Each repetition is one "iteration," and the results of one iteration are often used as the starting point for the next.
Here are 60 examples of iteration, explained across various fields.
๐ป Computer Science & Software
forLoop: A programming control structure that repeats a block of code a specific number of times. Explanation: It iterates over a sequence (like a list) or a range of numbers.whileLoop: A structure that repeats a block of code as long as a certain condition remains true. Explanation: It's used when you don't know the exact number of repetitions in advance.Array Processing: Reading or modifying each element in an array, one by one. Explanation: A loop iterates from the first element (index 0) to the last.
Gradient Descent: An algorithm in machine learning. Explanation: It iteratively adjusts a model's parameters to minimize a cost function (error), "descending" toward the best solution.
Agile Software Development: A project management methodology. Explanation: The project is built in a series of short, repeated cycles (called "sprints"), with each cycle adding and refining features.
Scrum: A specific Agile framework. Explanation: Work is processed in iterative "sprints," with a review and planning meeting after each one to set goals for the next.
Unit Testing: Running tests on a piece of code after making changes. Explanation: The "code-test-refactor" cycle is an iterative process of refinement.
Recursive Function (as iteration): A function that calls itself. Explanation: While technically "recursion," it solves a problem by applying the same logic repeatedly to smaller subproblems.
Rendering a 3D Scene: A computer graphics process. Explanation: The software iterates through pixels or objects, calculating lighting, shadows, and textures for each one.
Data Scraping: A script that browses a website to extract information. Explanation: It iterates through a list of URLs or pages, pulling data from each one.
๐ง Mathematics
Newton's Method: A technique for finding the roots (solutions) of an equation. Explanation: You start with a guess and repeatedly apply an iterative formula to get closer and closer to the actual answer.
Fibonacci Sequence: The sequence 0, 1, 1, 2, 3, 5, 8... Explanation: Each new number is generated by iterating on the previous step: adding the two preceding numbers.
Generating a Fractal (e.g., Mandelbrot Set): Creating a complex geometric shape. Explanation: A mathematical formula () is applied iteratively for each point on a graph.
Simple Interest Calculation (Year over Year): Calculating interest on the principal amount. Explanation: The same calculation () is repeated for each time period.
Compound Interest Calculation: Interest is added to the principal. Explanation: The calculation is iterated, but the starting principal (the input) changes with each cycle, as it now includes the interest from the last cycle.
Long Division: The standard arithmetic method. Explanation: You iterate through a process of "divide, multiply, subtract, bring down" for each digit.
Euclidean Algorithm: A method for finding the greatest common divisor of two numbers. Explanation: It's an iterative process where you repeatedly divide the larger number by the smaller one and take the remainder until the remainder is 0.
Convergent Sequence: A sequence that approaches a specific limit. Explanation: Each iteration (, , ...) gets progressively closer to a final value.
Markov Chain: A model of random events. Explanation: The probability of the next state is calculated iteratively, based only on the current state.
Power Iteration: An algorithm in linear algebra. Explanation: It's used to find the dominant eigenvector of a matrix by repeatedly multiplying the matrix by a vector.
๐งฌ Science & Engineering
The Scientific Method: The core process of scientific inquiry. Explanation: It's an iterative cycle: form a hypothesis, test it, analyze the results, and then refine the hypothesis for the next iteration.
Iterative Design Process (Engineering): A standard design methodology. Explanation: Engineers follow a "Plan-Do-Check-Act" or "Design-Prototype-Test-Refine" loop, with each iteration improving the product.
Evolution by Natural Selection: The biological process. Explanation: Each generation is an iteration. Favorable traits are "selected," become the input for the next generation, and are refined over time.
Iterative Evolution: A specific evolutionary phenomenon. Explanation: The same trait evolves independently in the same lineage at different points in time (e.g., the flightless Aldabra rail re-evolving after its ancestors went extinct).
Drug Development: Creating new medicines. Explanation: Scientists iteratively synthesize new molecules, test them, analyze the results, and then modify the molecular structure for the next round of testing.
Iterative Modeling (Systems Biology): Understanding complex biological systems. Explanation: A model (e.g., of a cell) is built, tested against real data, and then iteratively updated to better match reality.
Wind Tunnel Testing: Refining aerodynamics. Explanation: A model (of a car or plane) is tested, its design is tweaked, and then it's put back in the wind tunnel for the next iteration.
Protein Folding Simulation: A computational biology problem. Explanation: Computers iteratively test different configurations of a protein's structure to find the most stable (lowest energy) state.
Rocket Engine Design: Developing new engines (e.g., SpaceX's Raptor). Explanation: They build an engine, test it to failure, analyze the failure, build a new iteration, and repeat.
Climate Modeling: Predicting future climate. Explanation: A simulation is run in discrete time steps (e.g., every 10 minutes), with the output of one step (temperature, pressure) becoming the input for the next iteration.
๐จ Art & Creative Processes
Sketching a Drawing: The process of creating a final art piece. Explanation: You start with a rough "gesture" sketch, then iteratively refine it with more detail, clean lines, and shading.
Writing a Novel: The editing process. Explanation: A writer completes a first draft, then iterates on it through a second draft, a third draft, etc., refining plot and characters each time.
Songwriting: Crafting a song. Explanation: A musician might iterate on a melody, then the lyrics, then the harmony, constantly revisiting and tweaking each element.
Rehearsing a Play: Preparing for a performance. Explanation: Actors and the director run through scenes iteratively, making small adjustments to timing, delivery, and blocking each time.
Pottery on a Wheel: Shaping clay. Explanation: The potter iteratively pulls the clay upward, with each pass thinning and raising the walls of the pot.
Stop-Motion Animation: Creating an animated film. Explanation: The animator iterates a tiny action: "pose the model, take a picture, move the model slightly, take another picture."
Logo Design: A graphic design process. Explanation: A designer presents several concepts, gets feedback from the client, and then iterates on the chosen design to refine it.
Printmaking (e.g., Linocut): Creating multiple prints. Explanation: Each press of the paper onto the inked block is one iteration.
Musical Practice (Scales): A musician learning an instrument. Explanation: They repeat a scale or passage over and over, correcting notes and rhythm with each iteration.
Developing a Photograph (in a darkroom): A chemical process. Explanation: The photographer iteratively checks the print in the developer fluid, waiting for the image to reach the perfect exposure.
๐ถ Daily Life & Personal Development
Learning to Walk: A toddler's process. Explanation: Each attempt is an iteration of "stand, balance, step, fall, get up," with motor skills improving each time.
Cooking a Recipe: Refining a personal dish. Explanation: You make it once, then the next time you "iterate" by adding a little more garlic or less salt.
Studying with Flashcards: A learning technique. Explanation: You iterate through the deck, setting aside the ones you know and repeating the ones you don't.
Working Out (Progressive Overload): Building muscle. Explanation: You iterate your weekly routine, but with each new cycle, you slightly increase the weight or the number of reps.
Budgeting: Managing personal finances. Explanation: You create a budget for the month (iteration 1), and at the end of the month, you review and adjust it for the next month (iteration 2).
Commuting to Work: A daily routine. Explanation: You repeat the same sequence of steps (get in car, drive route, park) every day.
Washing Dishes: Cleaning a stack of dishes. Explanation: You iterate through the pile, repeating the "scrub, rinse, dry" process for each dish.
Setting an Alarm Clock: A daily event. Explanation: The alarm clock repeats its "ring" function at the same time in each 24-hour iteration (a day).
Checking Your Email: A common habit. Explanation: Many people iteratively "unlock phone, open app, scan for new messages, close app" throughout the day.
Breathing: A biological process. Explanation: Life is sustained by the constant, non-stop iteration of inhaling and exhaling.
๐️ Other Fields
Legal Precedent (Common Law): How law evolves. Explanation: A judge makes a ruling (iteration 1). A future judge in a similar case reviews that ruling and either follows it or refines it (iteration 2).
Product Prototyping (The Marshmallow Challenge): A design challenge. Explanation: Teams that build one tall tower and fail do worse than teams that iteratively "build a small structure, test it, build a slightly bigger one, test it."
Wiki Editing (e.g., Wikipedia): Collaborative content creation. Explanation: An article is published, then a user iterates on it by correcting a fact. Another user iterates by adding a new section.
Political Polling: Tracking public opinion. Explanation: A polling agency repeatedly asks the same questions to a new sample of people (e.g., every week) to see how opinions change.
Military Drill: Training soldiers. Explanation: A drill sergeant has a squad iterate a specific maneuver until it is performed perfectly and instinctively.
Restaurant Service: Running a restaurant. Explanation: The "take order, send to kitchen, deliver food, clear table" process is iterated for every customer.
Assembly Line (Manufacturing): Mass production. Explanation: A product moves down a line, and each station performs one iterative step on it (e.g., "add screw," "tighten screw," "inspect screw").
Supply Chain Management: Ordering inventory. Explanation: A store manager iteratively checks stock levels and places new orders to maintain inventory.
User Interface (UI) Design: Improving a website. Explanation: A company tracks user clicks, forms a hypothesis (e.g., "the button is too small"), makes an iterative change, and then measures the results.
Thomas Edison and the Lightbulb: A famous example of invention. Explanation: He didn't invent it in one try; he famously iterated through thousands of different filament materials until he found one that worked.
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Iteration is the process of repeating a set of instructions or an action, often to get closer to a goal, refine a result, or process a collection of items. Each individual repetition is called an "iteration."
Here are 60 examples of iteration, broken down by category.
## ๐ป 1. Computer Science & Programming
These are formal iterations used in algorithms and software.
1.
forLoop: Repeating code a specific number of times.Example: A program that loops 10 times to print the numbers 1 through 10.
2.
whileLoop: Repeating code as long as a specific condition remains true.Example: A game loop that continues to run
while (player_is_alive == true).
3.
do-whileLoop: Repeating code at least once, and then checking a condition to see if it should continue.Example: A program that asks a user for a password (at least once) and keeps asking while the password entry is incorrect.
4.
forEachLoop: Iterating over every single item in a collection (like a list or array).Example: A program that goes through a list of email addresses
['a@b.com', 'c@d.com']and sends an email to each one.
5. Recursion: A function that calls itself, repeating its own logic with a smaller or different piece of data.
Example: A function to calculate a factorial, where
factorial(5)callsfactorial(4), which callsfactorial(3), and so on.
6. Array Traversal: The general process of accessing each element in an array, one by one, from start to finish.
Example: Summing all the numbers in an array
[10, 20, 30]by adding10, then20, then30.
7. Map Operation: Applying the same function to every item in a list to create a new list.
Example: Taking
[1, 2, 3]and applying a "square" function to each item to produce[1, 4, 9].
8. Reduce/Fold Operation: Cumulatively combining all elements in a list into a single value.
Example: Summing
[1, 2, 3, 4]by performing(0+1) -> 1, then(1+2) -> 3, then(3+3) -> 6, then(6+4) -> 10.
9. Search Algorithm: Repeatedly checking parts of a dataset to find a specific value.
Example: A "binary search" repeatedly guessing the middle of a sorted list and discarding the half where the item isn't.
10. Sorting Algorithm: Repeatedly comparing and swapping pairs of elements in a list until the entire list is in order.
Example: "Bubble Sort" iterates through a list, swapping adjacent items if they are in the wrong order, and repeats this entire pass until no swaps are needed.
11. Data Processing Pipeline: A series of steps where the output of one step becomes the input for the next, repeated for all data.
Example: An ETL (Extract, Transform, Load) job that runs every night to fetch sales data, reformat it, and load it into a database.
12. Packet Processing: A network router iterating through its queue of incoming data packets, reading the destination of each one and forwarding it.
## ๐งช 2. Mathematics & Science
These are iterative methods used for calculation, modeling, and discovery.
13. The Scientific Method: The classic iterative process of
Hypothesis -> Test -> Analyze -> Refine Hypothesis.Example: A chemist's first test (iteration 1) fails, so they adjust the formula (iteration 2) and test again.
14. Newton's Method: An algorithm to find the root (or solution) of an equation by starting with a guess and applying an iterative formula to get a more and more accurate answer.
15. Monte Carlo Simulation: Running a computational model thousands of times, each time with different random inputs, to find the most probable outcomes.
Example: A financial model that simulates 10,000 possible stock market futures (10,000 iterations) to assess a portfolio's risk.
16. Fractal Generation: Creating a complex, self-similar pattern by applying a simple mathematical rule over and over.
Example: Drawing a Mandelbrot set by running the same equation for each pixel, iterating to see if the result "escapes" to infinity.
17. Numerical Integration: Approximating the area under a curve by summing the areas of many small shapes (like rectangles or trapezoids). Each new calculation with smaller shapes is a more accurate iteration.
18. Cellular Automata: A model of a grid where each "cell" changes state based on its neighbors. The entire grid is recalculated for each new "generation" (iteration).
Example: Conway's Game of Life, where patterns evolve over time in a repeating, step-by-step process.
19. Markov Chains: A model of states where the next step depends only on the current state.
Example: A weather model that iterates day-by-day, where the probability of "Rain" tomorrow depends only on whether today is "Sunny" or "Rainy."
20. Statistical "Bootstrapping": Repeatedly resampling from one's own dataset to estimate the uncertainty of a statistic (like the mean).
## ๐ 3. Everyday Life & Habits
These are common, often unconscious, repetitions.
21. Walking or Running: The repeated, cyclical motion of taking steps (left foot, right foot, left foot...).
22. Breathing: The continuous, rhythmic iteration of inhaling and exhaling.
23. Heartbeat: The repeated cycle of the heart muscle contracting (systole) and relaxing (diastole).
24. Following a Recipe: Any step that requires repetition.
Example: "Stir until combined" (a
whileloop) or "Knead the dough for 10 minutes" (aforloop).
25. Washing Dishes: Processing a stack of items one by one (pick up, scrub, rinse, place in rack) until the stack is empty.
26. Knitting or Crocheting: Repeating a specific stitch or pattern of stitches to create a row, and repeating rows to create a garment.
27. Building a Habit: Consciously repeating an action at the same time or cue each day.
Example: Meditating for 5 minutes every morning for 30 consecutive days (30 iterations).
28. Brushing Your Teeth: The back-and-forth scrubbing motion, repeated across different sections of your mouth.
29. Checking Your Phone: The (often compulsive) cycle of unlocking the phone, checking for notifications, and locking it again.
30. Commuting: Taking the same route to and from work every weekday. Each day's trip is one iteration.
31. Listening to a Song: The chorus repeating multiple times within the song, or putting the entire song "on loop."
32. Dealing Cards: A dealer handing out one card at a time to each player, repeating the cycle until all cards are dealt.
## ๐️ 4. Learning & Skill Development
These iterations are focused on practice and refinement.
33. Practicing an Instrument: Playing a difficult passage, scale, or song over and over to build muscle memory.
Example: A pianist playing a C-major scale 20 times in a row.
34. Studying Flashcards: Going through a deck of cards one by one, and then repeating the entire deck in future study sessions.
35. Sports Drills: Repeating a specific physical action to perfect it.
Example: A basketball player shooting 100 free throws, or a soccer player practicing penalty kicks.
36. Writing and Editing: The "writing, reviewing, revising" cycle.
Example: Writing a first draft (iteration 1), getting feedback and writing a second draft (iteration 2), and proofreading (iteration 3).
37. Memorizing a Speech: Reciting the speech from start to finish multiple times, making small corrections each time.
38. The "Code, Test, Debug" Cycle: The core loop for programmers: writing code, testing if it works, and fixing it (debugging) when it doesn't... then repeating the test.
39. Rehearsing a Play: The cast performing the entire play (an iteration), getting notes from the director, and then performing it again.
40. Reviewing Game Tapes: A sports team watching their last game (iteration 1), then watching their previous game (iteration 2) to find repeating patterns of error.
## ๐ญ 5. Engineering, Design & Manufacturing
These iterations are central to creating and improving products.
41. Agile Software Development: A project management system where teams build and release software in short, repeated cycles called "sprints."
Example: A 2-week sprint (iteration) to build a login page, followed by another 2-week sprint to build the user profile page.
42. Prototyping: The process of building a quick model (v1), testing it, gathering feedback, and then building an improved model (v2).
43. A/B Testing: An iterative experiment where two versions (A and B) of a webpage or app are shown to users. The "winning" version becomes the new baseline for the next A/B test.
44. Assembly Line: Each station on the line performs one repetitive task on every product that passes by.
Example: One worker attaches the left-front wheel to every car, all day.
45. Quality Assurance (QA) Testing: A tester repeatedly running a "test script" (a set of steps) on new versions of a product to ensure nothing is broken.
46. Machine Learning (Training): A model "learns" by processing a large dataset. Each pass through the entire dataset is called an "epoch" (one iteration of training).
47. Stress Testing: Repeatedly applying force or load to a physical object to find its failure point.
Example: A machine that bends a phone hinge back and forth 100,000 times.
48. Design Thinking: A popular problem-solving framework (Empathize, Define, Ideate, Prototype, Test) that is cyclical—the results from the "Test" phase are used to iterate and go back to the "Ideate" or "Prototype" phase.
## ๐ 6. Business & Processes
These are recurring cycles in organizational operations.
49. Budgeting Cycle: The monthly or quarterly process of reviewing past spending (iteration 1) to create the budget for the next period (iteration 2).
50. Inventory Auditing: Systematically counting every item in a warehouse, one by one, often in a repeating cycle (e.g., quarterly).
51. Performance Reviews: The recurring (e.g., annual) cycle of an employee and manager meeting to review performance and set goals for the next iteration.
52. The "OODA Loop": A decision-making cycle (Observe, Orient, Decide, Act). A business observes a competitor's move, orients to the new market, decides on a response, and acts... then immediately begins observing again.
53. Sales "Cold Calling": A salesperson going down a list of phone numbers and repeating the same script or sales pitch for each new person.
54. Scrum (Daily Stand-up): A daily (iterative) team meeting where each person answers the same three questions: "What did I do yesterday? What will I do today? What is blocking me?"
55. Filing Paperwork: A clerk performing the same set of actions (stamp, sign, file) for every form in a large stack.
## ๐ฆ 7. Nature & Biology
These are natural, cyclical iterations found in the physical and biological world.
56. Evolution by Natural Selection: The endless cycle of reproduction, genetic variation, and environmental selection, where each "generation" is an iteration on the species.
57. Cell Division (Mitosis): The process of a cell duplicating its components and splitting, a cycle that repeats to create growth.
58. Seasonal Cycles: The annual repetition of Spring, Summer, Autumn, and Winter, driven by the Earth's orbit.
59. Day/Night Cycle: The 24-hour iteration of light and dark, driven by the Earth's rotation.
60. Tidal Cycles: The repeated, predictable rise and fall of the ocean's water level, driven by the moon's gravitational pull.
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Of course. In statistical analysis, iteration is the process of repeating a calculation or procedure, with each new cycle (or "iteration") building on the previous one to get closer to a final, stable answer.
When applied to student test data, iteration is used in two primary ways:
As a human-driven process for improving instruction.
As a computational method inside statistical algorithms to find the most accurate results.
Here are key examples of how iteration is applied to the statistical analysis of student test data.
## 1. Iterative Analysis for Instruction (The "Human Loop")
This is the most common and practical application for educators. It's a cyclical process of analyzing data to make decisions, and then analyzing new data to see if those decisions worked.
Explanation: An educator or administrator doesn't just "analyze test scores" once. They use the results from one test to change their teaching, and then use the next test to measure the impact of that change. This is an iterative loop.
Example (Data-Driven Instruction):
Iteration 1 (Test 1): A 5th-grade teacher gives a math benchmark test in September. The data shows that 70% of students failed questions related to fractions.
Analysis & Action: The teacher forms a hypothesis: "My current method for teaching fractions isn't working." They spend the next three weeks re-teaching the concept using a new, hands-on method.
Iteration 2 (Test 2): The teacher gives a follow-up quiz on fractions. The new data shows that 85% of students are now proficient.
Analysis & Action: The teacher concludes the new method was effective. The next iteration might be to identify the 15% who are still struggling for small-group intervention.
## 2. Cluster Analysis (Finding "Hidden" Student Groups)
This is a data-mining technique used to automatically group students based on similar patterns in their test data. This is often done using the K-Means algorithm, which is purely iterative.
Explanation: Imagine you have scores for 100 students across reading, math, and science. You want to find natural "types" of students (e.g., "high-all-around," "strong-verbal/weak-math," "at-risk"). The K-Means algorithm iteratively refines these groups.
Example (K-Means Algorithm):
Start (Guess): The algorithm "guesses" 3 group centers (called centroids) on a scatterplot of the student scores.
Iteration 1: It assigns every student to the closest group center. It then recalculates the center of each group by finding the average score of all students assigned to it. The centers move.
Iteration 2: Because the centers moved, some students are now closer to a different group. The algorithm re-assigns those students and moves the centers again.
...Iterations 3, 4, 5...: This process of [re-assign students -> move centers] repeats over and over.
Finish: The algorithm stops when an iteration begins and no students change groups, meaning the group centers are stable. You are left with 3 well-defined clusters of students.
## 3. Item Response Theory (Finding "True" Ability & Question Difficulty)
This is the advanced statistical method that powers modern standardized tests (like the SAT, GRE, or state assessments). It uses iterative algorithms to solve a "chicken-and-egg" problem.
Explanation: To know a student's "true" ability, you need to know how hard the questions are. But to know how hard the questions are, you need to know the ability of the students who took them. How do you solve this? By iterating. The most common method is the Expectation-Maximization (EM) algorithm.
Example (EM Algorithm):
Start (Guess): The algorithm makes an initial guess at each student's ability (e.g., based on their raw score) or each question's difficulty.
Iteration 1 (E-Step): It uses the guessed question difficulties to calculate the expected ability of each student.
Iteration 1 (M-Step): It then uses those new student abilities to re-calculate (or maximize the likelihood of) the difficulty and discrimination of each question.
...Iterations 2, 3, ... 500...: It repeats this two-step "E-M" cycle. Each cycle, the estimates for both student ability and question parameters get slightly more accurate.
Finish: The algorithm stops (or "converges") when the estimates for ability and difficulty barely change from one iteration to the next.
## 4. Iterative Model Building (Predictive Regression)
This is used to build predictive models, such as identifying which students are at risk of failing a final exam or not graduating.
Explanation: You want to find the best formula to predict a student's final test score (e.g.,
Final_Score = ? * Attendance + ? * Homework_Avg + ? * Quiz_Score). Stepwise Regression is an iterative method to find the best combination of predictor variables.Example (Forward Stepwise Regression):
Iteration 1: The algorithm builds 3 simple models: (A)
Final_Scorepredicted byAttendance, (B)Final_ScorebyHomework_Avg, and (C)Final_ScorebyQuiz_Score. It finds that model (C) is the best single predictor.Iteration 2: It keeps
Quiz_Scoreand tries adding the next-best variable. It tests: (A)Quiz_Score + Attendanceand (B)Quiz_Score + Homework_Avg. It finds that model (A) is the strongest combination.Iteration 3: It keeps
Quiz_Score + Attendanceand checks if addingHomework_Avgsignificantly improves the model. If it doesn't, the algorithm stops.Finish: The final model is the one built iteratively that provides the most predictive power without being overly complex.
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Here are examples of
if-then-elsestatements as they would appear in several common programming languages.## ๐ Python
Python is known for its clean syntax that uses indentation (whitespace) instead of curly braces to define code blocks. It also uses
elifas a shortcut for "else if."Example 1: Basic
ifThis block only runs if the condition is true.Pythonage = 20 if age >= 18: print("You are eligible to vote.")Output:
You are eligible to vote.Example 2:
if-elseOne of the two blocks will always run.Pythontemperature = 15 if temperature > 25: print("It's a hot day.") else: print("It's not a hot day.")Output:
It's not a hot day.Example 3:
if-elif-elseThis checks multiple conditions in order. It stops as soon as it finds one that is true.Pythongrade = 85 if grade >= 90: print("You got an A.") elif grade >= 80: print("You got a B.") elif grade >= 70: print("You got a C.") else: print("You need to study more.")Output:
You got a B.## JavaScript (and other C-style languages)
JavaScript, C++, C#, and Java all use a similar "C-style" syntax with curly braces
{}to define code blocks and parentheses()around the condition.Example 1: Basic
ifJavaScriptlet age = 20; if (age >= 18) { console.log("You are eligible to vote."); }Output:
You are eligible to vote.Example 2:
if-elseJavaScriptlet temperature = 15; if (temperature > 25) { console.log("It's a hot day."); } else { console.log("It's not a hot day."); }Output:
It's not a hot day.Example 3:
if-else if-elseNote the use ofelse if(two separate words).JavaScriptlet grade = 85; if (grade >= 90) { console.log("You got an A."); } else if (grade >= 80) { console.log("You got a B."); } else if (grade >= 70) { console.log("You got a C."); } else { console.log("You need to study more."); }Output:
You got a B.
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