What is the ratio of the sums of the first (64) and first (32) natural numbers?
Answer and explanation
Correct answer: \(130:33\)
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{64}=\frac{64\times65}{2}=2080\) and \(S_{32}=\frac{32\times33}{2}=528\). Therefore, \(S_{64}:S_{32}=2080:528=130:33\). The ratio \(65:33\) is incorrect because 2080 and 528 cannot both be divided by 32. Exam tip: always reduce a ratio using the greatest common factor of its two terms.
Frequently asked questions
What is the correct answer to this question?
\(130:33\)
Why is this the correct answer?
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{64}=\frac{64\times65}{2}=2080\) and \(S_{32}=\frac{32\times33}{2}=528\). Therefore, \(S_{64}:S_{32}=2080:528=130:33\). The ratio \(65:33\) is incorrect because 2080 and 528 cannot both be divided by 32. Exam tip: always reduce a ratio using the greatest common factor of its two terms.
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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