Sequences And Series — Medium JEE math MCQ
The sum of the series \(1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots + 10 \cdot 2^9\) is:
- A. \(9 \cdot 2^{10} + 1\)
- B. \(10 \cdot 2^{10} - 1\)
- C. \(9 \cdot 2^{10} - 1\)
- D. \(8 \cdot 2^{10} + 1\)
Solution
Let \(S = 1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots + 10 \cdot 2^9\)
Multiply by 2:
\(2S = 1 \cdot 2 + 2 \cdot 2^2 + 3 \cdot 2^3 + \ldots + 10 \cdot 2^{10}\)
Subtract:
\[
S - 2S = 1 + 2 + 2^2 + 2^3 + \ldots + 2^9 - 10 \cdot 2^{10}
\]
\[
-S = (2^{10} - 1) - 10 \cdot 2^{10} = 2^{10} - 1 - 10 \cdot 2^{10} = -9 \cdot 2^{10} - 1
\]
\[
S = 9 \cdot 2^{10} + 1
\]
So the correct option is A.