1 + 3 + 5 + 7 + ... + 17 is equal to ________.

Question

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Sum of Odd Numbers Theorem):","type":"text"},{"text":" The sum of the first ","type":"text"},{"attrs":{"latex":"n"},"type":"math_inline"},{"text":" positive odd integers is always equal to ","type":"text"},{"attrs":{"latex":"n^2"},"type":"math_inline"},{"text":":","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"1 + 3 + 5 + \dots + (2n-1) = n^2"},"type":"math_block"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 2 (Determine the number of terms n):","type":"text"},{"text":" The last term in our sum is ","type":"text"},{"attrs":{"latex":"17"},"type":"math_inline"},{"text":". We set up the relation:","type":"text"}]},{"attrs":{"latex":"2n - 1 = 17"},"type":"math_block"},{"attrs":{"latex":"2n = 18 \implies n = 9"},"type":"math_block"},{"content":[{"text":"So there are exactly 9 terms in the summation.","type":"text"}],"type":"paragraph"},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 3 (Apply the Sum Formula):"}],"type":"paragraph"},{"attrs":{"latex":"\text{Sum} = 9^2 = 81"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4 (Conclusion):","type":"text"},{"text":" The sum is exactly ","type":"text"},{"marks":[{"type":"code"}],"text":"81","type":"text"},{"text":".","type":"text"}],"type":"paragraph"}],"type":"doc"}

Guide / Hint

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