What is the ratio of the sums of the first (45) and first (30) natural numbers?
Answer and explanation
Correct answer: 69:31
The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{45}=\frac{45\times46}{2}=1035\) and \(S_{30}=\frac{30\times31}{2}=465\). Therefore, \(S_{45}:S_{30}=1035:465=69:31\). The option \(45:30\) is only the ratio of the numbers of terms, not the ratio of their sums. Exam tip: For ratios of such sums, apply \(\frac{n(n+1)}{2}\) first and then cancel common factors.
Frequently asked questions
What is the correct answer to this question?
69:31
Why is this the correct answer?
The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{45}=\frac{45\times46}{2}=1035\) and \(S_{30}=\frac{30\times31}{2}=465\). Therefore, \(S_{45}:S_{30}=1035:465=69:31\). The option \(45:30\) is only the ratio of the numbers of terms, not the ratio of their sums. Exam tip: For ratios of such sums, apply \(\frac{n(n+1)}{2}\) first and then cancel common factors.
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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