If the sum of the first n natural numbers is 465, what is n?
Answer and explanation
Correct answer: 30
The first n natural numbers are 1, 2, 3, ..., n, an arithmetic progression with first term 1 and last term n. Its sum is S_n = n(n + 1)/2. Equating this to 465 gives n(n + 1)/2 = 465, so n(n + 1) = 930. Since 30 × 31 = 930, n = 30. The positive value is selected because n counts terms and must be a positive integer. Therefore, option C is correct. The nearby choices 28, 29, and 31 fail the equation: their products with their successors are 812, 870, and 992, respectively, not 930.
Frequently asked questions
What is the correct answer to this question?
30
Why is this the correct answer?
The first n natural numbers are 1, 2, 3, ..., n, an arithmetic progression with first term 1 and last term n. Its sum is S_n = n(n + 1)/2. Equating this to 465 gives n(n + 1)/2 = 465, so n(n + 1) = 930. Since 30 × 31 = 930, n = 30. The positive value is selected because n counts terms and must be a positive integer. Therefore, option C is correct. The nearby choices 28, 29, and 31 fail the equation: their products with their successors are 812, 870, and 992, respectively, not 930.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first $n$ terms of an AP.
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