What is the ratio of the sums of the first (150) and first (100) natural numbers?
Answer and explanation
Correct answer: \(453:202\)
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{150}=\frac{150\times151}{2}=11325\) and \(S_{100}=\frac{100\times101}{2}=5050\). Hence, \(S_{150}:S_{100}=11325:5050=453:202\). \(3:2\) is only the ratio of 150 and 100; the factors \(151\) and \(101\) must also be included for the sums. Exam tip: In 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?
\(453:202\)
Why is this the correct answer?
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{150}=\frac{150\times151}{2}=11325\) and \(S_{100}=\frac{100\times101}{2}=5050\). Hence, \(S_{150}:S_{100}=11325:5050=453:202\). \(3:2\) is only the ratio of 150 and 100; the factors \(151\) and \(101\) must also be included for the sums. Exam tip: In 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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