If S_n - S_{n-1} = 84 and S_m = 990, what is n - m?
Answer and explanation
Correct answer: 40
The governing property is that consecutive partial sums of the natural-number sequence differ by the newly added term: S_n - S_{n-1} = n. Thus the first condition gives n = 84. For the second condition, S_m = m(m+1)/2 = 990, so m(m+1) = 1980. The positive integer solution is m = 44, because 44 × 45 / 2 = 990. Consequently, n - m = 84 - 44 = 40. Therefore option C is correct. The nearby values 38, 39, and 41 would come from mis-solving the quadratic sum equation or subtracting an incorrect index.
Frequently asked questions
What is the correct answer to this question?
40
Why is this the correct answer?
The governing property is that consecutive partial sums of the natural-number sequence differ by the newly added term: S_n - S_{n-1} = n. Thus the first condition gives n = 84. For the second condition, S_m = m(m+1)/2 = 990, so m(m+1) = 1980. The positive integer solution is m = 44, because 44 × 45 / 2 = 990. Consequently, n - m = 84 - 44 = 40. Therefore option C is correct. The nearby values 38, 39, and 41 would come from mis-solving the quadratic sum equation or subtracting an incorrect index.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Sum of first n natural numbers.
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