Sum a repeating number sequence

4.OA.C.53.OA.A.3 · take · grade 4

Archetype: Sum of Evenly Spaced Numbers via the Middle · step in a 5-type progression

The numbers below are arranged according to a rule. Find the repeating block, then find the sum of the numbers when they are written out to the $15$ th term.

$1,\ 3,\ 8,\ 1,\ 3,\ 8,\ 1,\ 3,\ \ldots$

Show solution

Understand

The list 1, 3, 8, 1, 3, 8, 1, 3, ... repeats a block. Find the repeating block, then add up the first 15 terms.

Givens

The sequence is 1, 3, 8, 1, 3, 8, 1, 3, ...
The same block of numbers repeats over and over.

Unknowns

The sum of the first 15 terms of the sequence.

Constraints

The repeating block is 1, 3, 8 (length 3).
We need exactly the first 15 terms.

Plan

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

The list repeats the block 1, 3, 8, so I find the block, count how many whole blocks fit into 15 terms (an easier sub-count), then add up the block sums.

Execute

#5 Look for a Pattern 4.OA.C.5

The numbers repeat in groups of three: 1, 3, 8. One block sums to 1 + 3 + 8 = 12.

1 + 3 + 8 = 12

Spotting that the same three numbers come back tells you the block and its repeat length.

#9 Solve an Easier Related Problem 3.OA.A.3

Each block has 3 terms, and 15 terms divided into groups of 3 gives 5 complete blocks with none left over.

15 \div 3 = 5

Because 15 is a multiple of 3, the 15 terms split evenly into 5 whole blocks.

#5 Look for a Pattern 3.OA.A.1

Five blocks, each summing to 12, give a total of 5 x 12 = 60.

5 \times 12 = 60

Repeated equal groups of 12 are quickest added by multiplying.

Answer: 60

Review

15 terms averaging 4 each (the block average is 12/3 = 4) would total about 60, which matches exactly. Since 15 is a clean multiple of 3, no partial block is left to worry about.

You could write out all 15 terms (1,3,8 five times) and add them one by one to confirm the running total reaches 60.

Standards · min grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Recognizing the repeating block 1, 3, 8 in the sequence.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing 15 terms into groups of 3 to count the blocks.
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups — Multiplying 5 blocks by the block sum 12 to get the total.

💡 Find the repeating block, see how many fit, and multiply: 5 blocks of (1+3+8) is just 5 x 12 = 60!