Sum of first n natural numbers

Correct answer: C. (28441)

The direct answer is C, 28441. For the sum of the first n natural numbers, use the formula \\(S_n=\\frac{n(n+1)}{2}\\). We are given \\(S_n=31375\\). First find n: \\(\\frac{250\\times251}{2}=125\\times251=31375\\), so n=250. The question asks for \\(S_{n-12}\\), so \\(n-12=250-12=238\\). Now calculate: \\(S_{238}=\\frac{238\\times239}{2}=119\\times239=28441\\). Therefore option C is correct. Option A, 28100, is not the value produced by the required sum formula. Option B, 28270, is also an incorrect calculation. Option C matches the exact step-by-step result. Option D, 28613, is another distractor and does not equal \\(\\frac{238\\times239}{2}\\). A common mistake is to subtract 12 from the sum instead of first finding n and then subtracting 12 from the index. Always identify the index before applying the sum formula.

Correct answer: A. \(8S_n+1\)

Using \(S_n=\frac{n(n+1)}{2}\), we get \(8S_n+1=4n(n+1)+1=(2n+1)^2\), which is always an odd perfect square. Exam tip: substitute the sum formula and look for a perfect-square identity.