What is the ratio of the sums of the first (84) and first (56) natural numbers?
Answer and explanation
Correct answer: \(85:38\)
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{84}=\frac{84\times85}{2}=3570\) and \(S_{56}=\frac{56\times57}{2}=1596\). Therefore, \(3570:1596=85:38\), in simplest form. \(84:56=3:2\) is only the ratio of the number of terms, not the ratio of their sums. Exam tip: For ratios of such sums, apply \(\frac{n(n+1)}{2}\) to both values of \(n\).
Frequently asked questions
What is the correct answer to this question?
\(85:38\)
Why is this the correct answer?
The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{84}=\frac{84\times85}{2}=3570\) and \(S_{56}=\frac{56\times57}{2}=1596\). Therefore, \(3570:1596=85:38\), in simplest form. \(84:56=3:2\) is only the ratio of the number of terms, not the ratio of their sums. Exam tip: For ratios of such sums, apply \(\frac{n(n+1)}{2}\) to both values of \(n\).
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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