Question:
A series \(S\) is given as \(S=1+3+5+7+9+\dots\). The sum of the first 50 terms of \(S\) is __________________.
A series \(S\) is given as \(S=1+3+5+7+9+\dots\). The sum of the first 50 terms of \(S\) is __________________.
Show Hint
- Odd numbers form an AP with \(a=1,\,d=2\).
- Memorize \(1+3+\cdots+(2n-1)=n^2\) \(\Rightarrow\) here \(n=50\Rightarrow 50^2=2500\).
- Memorize \(1+3+\cdots+(2n-1)=n^2\) \(\Rightarrow\) here \(n=50\Rightarrow 50^2=2500\).
Updated On: Aug 26, 2025
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Correct Answer: 2500
Solution and Explanation
Step 1: This is an AP with first term \(a=1\) and common difference \(d=2\).
Step 2: Sum of first \(n\) terms: \(S_n=\dfrac{n}{2}\,[2a+(n-1)d]\).
Step 3: For \(n=50\): \(S_{50}=\dfrac{50}{2}[2\cdot1+49\cdot2]=25(2+98)=25\cdot100=2500\).
Step 2: Sum of first \(n\) terms: \(S_n=\dfrac{n}{2}\,[2a+(n-1)d]\).
Step 3: For \(n=50\): \(S_{50}=\dfrac{50}{2}[2\cdot1+49\cdot2]=25(2+98)=25\cdot100=2500\).
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