Problem
Sum a repeating number sequence
The list 1, 3, 8, 1, 3, 8, 1, 3, ... repeats a block. Find the repeating block, then add up the first 15 terms.
Operations
Your answer
Watch out This asks for the sum of all 15 terms, not the sum of just one block.
How to solve
Strategy Look for a Pattern — 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.
1STEP 1
Identify the repeating block and its sum
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.
4.OA.C.5Look For A Pattern2STEP 2
Count how many blocks make 15 terms
Each block has 3 terms, and 15 terms split into groups of 3 give 5 complete blocks with none left over.
15 ÷ 3 = 5
Because 15 is a multiple of 3, the 15 terms split evenly into 5 whole blocks.
3.OA.A.3Solve An Easier Related Problem3STEP 3
Add the block sums
Five blocks, each summing to 12, give a total of 5 x 12 = 60.
5 × 12 = 60
Repeated equal groups of 12 are quickest added by multiplying.
3.OA.A.1Look For A PatternAdding up the first 15 terms gives the same total as five equal groups of the block sum, 12.
Why?
The first 15 terms are exactly five copies of the block 1, 3, 8, so the long sum breaks into five block-sums added together.
Why?
The rule repeats the same block 1, 3, 8 without stopping, and 15 terms is five whole lengths of that three-term block.
Why?
Counting the 15 positions three at a time fills exactly five groups, since five groups of three make fifteen.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The 15 terms are cut into blocks with no term left out and none counted twice, so the five block-sums add back to the total of all 15.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Every block holds the same numbers 1, 3, 8, so each block-sum is the same amount, and adding that one amount five times is what multiplying by five means.
Why?
Five equal block-sums added together is the same as five groups of that amount, which is exactly five times the amount.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
60
5 × 12 = 60
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.
Takeaway
Find the repeating block, see how many fit, and multiply: 5 blocks of (1+3+8) is just 5 x 12 = 60!
- Identify the repeating block and its sum
- Count how many blocks make 15 terms
- Add the block sums
Where next?
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