What is the ratio of the sums of the first (128) and first (64) natural numbers?
Answer and explanation
Correct answer: \(258:65\)
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{128}=\frac{128\times129}{2}=8256\) and \(S_{64}=\frac{64\times65}{2}=2080\). Therefore, \(8256:2080=258:65\), so the correct answer is \(258:65\). \(2:1\) is only an approximate comparison, not the exact simplest ratio. Exam tip: write the sum formula for both terms first, then cancel common factors in the ratio.
Frequently asked questions
What is the correct answer to this question?
\(258:65\)
Why is this the correct answer?
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{128}=\frac{128\times129}{2}=8256\) and \(S_{64}=\frac{64\times65}{2}=2080\). Therefore, \(8256:2080=258:65\), so the correct answer is \(258:65\). \(2:1\) is only an approximate comparison, not the exact simplest ratio. Exam tip: write the sum formula for both terms first, then cancel common factors in the ratio.
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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