5Question: What is the remainder when the sum of the first 10 terms of the arithmetic sequence starting at 1 with common difference 4 is divided by 9?

February 22, 2026

5Question: What is the remainder when the sum of the first 10 terms of the arithmetic sequence starting at 1 with common difference 4 is divided by 9?

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Solution: The sequence is $1, 5, 9, 13, 17, 21, 25, 29, 33, 37$. The sum of an arithmetic sequence is given by $ S = \frac{n}{2}(2a + (n-1)d) $. Here, $ n = 10 $, $ a = 1 $, $ d = 4 $.
S = \frac{10}{2}(2 \cdot 1 + 9 \cdot 4) = 5(2 + 36) = 5 \cdot 38 = 190
Now, compute $ 190 \mod 9 $. Since $ 9 \cdot 21 = 189 $, the remainder is $ 190 - 189 = 1 $.

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