If the sum of the first n even natural numbers is 420, what is the value of n?
Answer and explanation
Correct answer: 20
The first n even natural numbers are 2, 4, 6, ..., 2n. Their sum is 2(1 + 2 + ... + n) = 2[n(n + 1)/2] = n(n + 1). Therefore n(n + 1) = 420. We look for consecutive positive integers whose product is 420: 20 × 21 = 420. Hence n = 20, so option C is correct. Checking directly, the first 20 even numbers end at 40 and their sum is 20 × 21 = 420. The other choices give products 342, 380, and 462, so they do not satisfy the condition.
Frequently asked questions
What is the correct answer to this question?
20
Why is this the correct answer?
The first n even natural numbers are 2, 4, 6, ..., 2n. Their sum is 2(1 + 2 + ... + n) = 2[n(n + 1)/2] = n(n + 1). Therefore n(n + 1) = 420. We look for consecutive positive integers whose product is 420: 20 × 21 = 420. Hence n = 20, so option C is correct. Checking directly, the first 20 even numbers end at 40 and their sum is 20 × 21 = 420. The other choices give products 342, 380, and 462, so they do not satisfy the condition.
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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