Repeated Operations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Repeated Operations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Applying the same operation multiple times in succession, where the repetition is often compressed into a higher-level operation: repeated addition becomes multiplication ( $n \cdot a$ ), and repeated multiplication becomes exponentiation ( $a^n$ ).

Adding 5 three times: $5+5+5 = 3 \times 5$ . Multiplying 2 four times: $2 \times 2 \times 2 \times 2 = 2^4$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Repeating an operation collapses into a higher one: many equal additions become a multiplication, many equal multiplications become a power.

Common stuck point: The procedure for repeated operations is the easy part; the trap is compressing unequal terms. Asking "Is the identical operation applied to the same number several times in a row?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the identical operation applied to the same number several times in a row?

Worked Examples

Example 1

easy

Start at 3 and add 4 repeatedly. Write the first 5 results.

Answer

3, 7, 11, 15, 19

First step

Start: 3.

Full solution

2
After 1st addition: $3 + 4 = 7$ .
3
After 2nd: $7 + 4 = 11$ .
4
After 3rd: $11 + 4 = 15$ .
5
After 4th: $15 + 4 = 19$ .

Repeated addition of a constant creates an arithmetic sequence. Each term is 4 more than the previous.

Example 2

medium

Start with 2 and repeatedly double it. What are the first 5 values? What operation are you applying each time?

Example 3

easy

Start at 12 and subtract 3 repeatedly. Write the first 5 results.

Example 4

medium

Start with 1 and multiply by 4 repeatedly. What are the first 5 values?

Example 5

medium

Compare: adding 4 fifteen times vs computing

15\times 4

. Which is faster, and what is the value?

Example 6

hard

Compute

4+4+4+4+4+4+4+4+4+4+4+4

two ways: as a sum and as

n\times a

. Confirm both give the same value.

Example 7

hard

Express the sum

5+5+5+5+5+5+5+5+5+5

first as

n\times a

, then as a power involving 5 if possible.

Example 8

challenge

A grain of rice is placed on a chessboard such that each square has double the grains of the previous one, starting with 1 grain on square 1. How many grains are on square 11?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Start at 50 and subtract 7 repeatedly. Write the first 4 results.

Example 2

medium

Start with 1 and multiply by 3 repeatedly. What is the 5th value in the sequence?

Example 3

easy

Write

4+4+4+4+4

as a multiplication.

Example 4

easy

Write

2\times 2\times 2

as a power.

Example 5

easy

Compute

3\times 6

as repeated addition.

Example 6

easy

What is

5^2

Example 7

easy

How many times is 7 added in

4\times 7

Example 8

easy

Write

10+10+10

as a multiplication and find the value.

Example 9

easy

What is

3^3

Example 10

easy

Express '6 added 4 times' and '4 added 6 times' — are the totals equal?

Example 11

medium

A pattern adds 3 each step starting at 5:

5,8,11,\dots

. Write the value of step

n

Example 12

medium

Rewrite

2^5

as a product and evaluate.

Example 13

medium

Which grows faster after 5 steps: adding 3 each time from 0, or multiplying by 3 each time from 1?

Example 14

medium

A bacteria doubles every hour starting at 4. How many after 3 hours?

Example 15

medium

Simplify

7+7+7+x+x

using repeated-operation shorthand.

Example 16

medium

How many factors of 5 are in

5^4

, and what is its value?

Example 17

medium

A staircase pattern uses

1,3,5,7,\dots

blocks per step. How many blocks in step 6?

Example 18

medium

Express the total of

6+6+6+6+6+6+6

two ways and evaluate.

Example 19

challenge

A pattern triples each step:

2,6,18,\dots

. Find a formula for step

n

and the value at step 5.

Example 20

challenge

Show that adding 5 a total of

k

times equals

5k

, then find

k

if the total is 60.

Example 21

challenge

2^n = 64

, find

n

by recognizing repeated multiplication.

Example 22

medium

Write

9+9+9+9

as a product and as a multiple, then evaluate.

Example 23

easy

Write

7+7+7+7+7+7

as a multiplication and find the value.

Example 24

easy

Rewrite

5\times 5\times 5\times 5

as a power.

Example 25

easy

How many times is 8 added in

6\times 8

Example 26

medium

A jar gets 5 marbles added each day. After 12 days, how many marbles have been added?

Example 27

medium

A colony of cells doubles every hour. Starting with 1 cell, how many cells are there after 6 hours?

Example 28

medium

Rewrite

a+a+a+a+a+a+a

using multiplication.

Example 29

medium

Rewrite

x\cdot x\cdot x\cdot x\cdot x

as a power.

Example 30

medium

A bacterium triples every minute. After 4 minutes, by what factor has the count grown?

Example 31

medium

Write

\underbrace{2+2+\cdots+2}_{20 \text{ times}}

as a product.

Example 32

medium

Write

\underbrace{10\cdot 10\cdots 10}_{6 \text{ times}}

as a power and compute.

Example 33

hard

Aiden saves \$6 each week for 1 year (52 weeks). How much does he save? Express both as repeated addition and as a product.

Example 34

hard

A piece of paper is folded in half 8 times. How many layers are there after the last fold?

Example 35

hard

How many times must you multiply 2 by itself to reach 128?

Example 36

hard

A shop owner triples her inventory every month. Starting with 5 items, how many items does she have after 3 months?

Example 37

hard

Compute

2^{10}

Example 38

challenge

A snail climbs 3 ft each day and slips back 1 ft each night, then repeats. After 10 full day-night cycles, how far up is it?

← Back to Repeated Operations Formula Explained Practice Mode

Related Concepts

Addition Multiplication Exponents Sequence

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionmultiplication