If S_n = 1830, what is the value of S_{n+10} - S_n?
Answer and explanation
Correct answer: 655
For the sum of the first n natural numbers, S_n = 1 + 2 + ... + n = n(n+1)/2. Since n(n+1)/2 = 1830, we get n(n+1) = 3660, so n = 60. Therefore, S_{n+10} - S_n = S_70 - S_60, which contains only the ten new terms 61 through 70. Their sum is an arithmetic progression: 10/2 × (61 + 70) = 5 × 131 = 655. Hence option B is correct. The other choices result from an incorrect endpoint or an arithmetic addition error; the given value of S_n is used first to identify n.
Frequently asked questions
What is the correct answer to this question?
655
Why is this the correct answer?
For the sum of the first n natural numbers, S_n = 1 + 2 + ... + n = n(n+1)/2. Since n(n+1)/2 = 1830, we get n(n+1) = 3660, so n = 60. Therefore, S_{n+10} - S_n = S_70 - S_60, which contains only the ten new terms 61 through 70. Their sum is an arithmetic progression: 10/2 × (61 + 70) = 5 × 131 = 655. Hence option B is correct. The other choices result from an incorrect endpoint or an arithmetic addition error; the given value of S_n is used first to identify n.
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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