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Sum of 3+5+9+17+33+... to n Terms | JEE

JEE Maths question with a full step-by-step solution.

Question
The sum of the series 3+5+9+17+33+⋯3+5+9+17+33+\cdots to nn terms is
A2n+1−n−22^{n+1}-n-2
B2n+1+n−22^{n+1}+n-2correct
C2n+n−22^n+n-2
D2n+1−n+22^{n+1}-n+2
Solution
Step 1: Each term is a power of 22 plus 11:
Sn=(2+1)+(22+1)+(23+1)+⋯+(2n+1)=(2+22+⋯+2n)+n.S_n=(2+1)+(2^2+1)+(2^3+1)+\cdots+(2^n+1)=(2+2^2+\cdots+2^n)+n.
 Step 2:
Sn=2(2n−1)+n=2n+1+n−2.S_n=2(2^n-1)+n=2^{n+1}+n-2.
 Correct answer: (2)
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