What is the ratio of (S_{90}) to (S_{45})?
Answer and explanation
Correct answer: 91:23
The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{90}=\frac{90\times91}{2}=4095\) and \(S_{45}=\frac{45\times46}{2}=1035\). Hence, \(S_{90}:S_{45}=4095:1035=91:23\). The option \(90:23\) fails to account correctly for the \(n+1\) factor. Exam tip: for such ratios, write the sum formula first and cancel common factors before calculating fully.
Frequently asked questions
What is the correct answer to this question?
91:23
Why is this the correct answer?
The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{90}=\frac{90\times91}{2}=4095\) and \(S_{45}=\frac{45\times46}{2}=1035\). Hence, \(S_{90}:S_{45}=4095:1035=91:23\). The option \(90:23\) fails to account correctly for the \(n+1\) factor. Exam tip: for such ratios, write the sum formula first and cancel common factors before calculating fully.
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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