If the sum of the first n natural numbers is 2701, what is the value of n?
Answer and explanation
Correct answer: 73
The governing formula for the sum of the first n natural numbers is Sₙ = n(n + 1)/2. We need n(n + 1)/2 = 2701, or n(n + 1) = 5402. Testing the nearby choices is efficient: for n = 73, the product is 73 × 74 = 5402, and therefore S₇₃ = 73 × 74/2 = 73 × 37 = 2701. Hence n = 73 and option C is correct. The neighboring values do not work: S₇₂ = 72 × 73/2 = 2628, while S₇₄ = 74 × 75/2 = 2775. Since the sum increases strictly as n increases, only one option can satisfy the equation.
Frequently asked questions
What is the correct answer to this question?
73
Why is this the correct answer?
The governing formula for the sum of the first n natural numbers is Sₙ = n(n + 1)/2. We need n(n + 1)/2 = 2701, or n(n + 1) = 5402. Testing the nearby choices is efficient: for n = 73, the product is 73 × 74 = 5402, and therefore S₇₃ = 73 × 74/2 = 73 × 37 = 2701. Hence n = 73 and option C is correct. The neighboring values do not work: S₇₂ = 72 × 73/2 = 2628, while S₇₄ = 74 × 75/2 = 2775. Since the sum increases strictly as n increases, only one option can satisfy the equation.
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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