Problem
Sum a repeating number sequence
The numbers below are arranged by a rule: 1, 3, 8, 1, 3, 8, 1, 3, …. Find the repeating block, then find the sum of the numbers when they are written out to the 15th term.
Operations
Your answer
Watch out The answer isn't just one block's sum of 12 — it's all 5 whole blocks that make up the 15 terms, added together.
How to solve
Strategy Look for a Pattern — The sequence repeats the block 1, 3, 8, so find the block, count how many whole blocks fit into 15 terms, then multiply by the block sum.
1STEP 1
Identify the repeating block and its sum
The numbers repeat in blocks of 1, 3, 8. One block sums to 1+3+8=12.
1 + 3 + 8 = 12
Seeing the same three numbers 1, 3, 8 come back tells you the block length is 3, and adding them gives one block's sum, 12.
Look For A Pattern4.OA.C.5Finding Patterns2STEP 2
Count how many blocks make 15 terms
Each block has 3 terms, so 15 ÷ 3 = 5 whole blocks fit with none left over.
15 ÷ 3 = 5
Because 15 is a multiple of 3, grouping the terms three at a time gives exactly 5 groups with nothing left over, so the 15 terms split evenly into 5 blocks.
Solve An Easier Related Problem3.OA.A.3Finding Patterns3STEP 3
Add the block sums
Five blocks each summing to 12 give 5 × 12 = 60.
5 × 12 = 60
The 15 terms are exactly five copies of the block, so the total is five block-sums added together, and since every block-sum is the same 12, adding that amount five times is the same as multiplying by five.
Look For A Pattern3.OA.A.1Finding PatternsAdding 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
Check — writing 1, 3, 8 five times gives fifteen terms, and adding them one by one also totals 60.
Takeaway
Find the sum of the repeating block, count how many blocks fit, then multiply to get the total.
- Block 1, 3, 8 sums to 12
- 15 ÷ 3 = 5 blocks
- 5 × 12 = 60
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