Question:
Sum of the series to \(n\) terms \(5 + 7 + 13 + 31 + 85 + \ldots\) is
Sum of the series to \(n\) terms \(5 + 7 + 13 + 31 + 85 + \ldots\) is
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When second differences form a GP, \(T_n = A \cdot r^{n-1} + B\).
Updated On: Apr 23, 2026
- \(3^n + 8n + 1\)
- \(\frac{1}{2}\left[3^n + 8n - 1\right]\)
- \(\frac{1}{2}\left(3^n + 8n + 1\right)\)
- None of the above
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The Correct Option is B
Solution and Explanation
Step 1: Formula / Definition}
\[ T_n = 3^{n-1} + 4 \]
Step 2: Calculation / Simplification}
Differences: \(7-5=2\), \(13-7=6\), \(31-13=18\), \(85-31=54\)
Second differences: \(6-2=4\), \(18-6=12\), \(54-18=36\) (GP with ratio 3)
\[ T_n = A \cdot 3^{n-1} + B \]
\(n=1: A+B=5\); \(n=2: 3A+B=7\) \(\Rightarrow A=1, B=4\)
\[ S_n = \sum_{k=1}^n (3^{k-1} + 4) = \frac{3^n - 1}{3-1} + 4n = \frac{3^n - 1}{2} + 4n \]
\[ = \frac{3^n + 8n - 1}{2} \]
Step 3: Final Answer
\[ \frac{1}{2}\left[3^n + 8n - 1\right] \]
\[ T_n = 3^{n-1} + 4 \]
Step 2: Calculation / Simplification}
Differences: \(7-5=2\), \(13-7=6\), \(31-13=18\), \(85-31=54\)
Second differences: \(6-2=4\), \(18-6=12\), \(54-18=36\) (GP with ratio 3)
\[ T_n = A \cdot 3^{n-1} + B \]
\(n=1: A+B=5\); \(n=2: 3A+B=7\) \(\Rightarrow A=1, B=4\)
\[ S_n = \sum_{k=1}^n (3^{k-1} + 4) = \frac{3^n - 1}{3-1} + 4n = \frac{3^n - 1}{2} + 4n \]
\[ = \frac{3^n + 8n - 1}{2} \]
Step 3: Final Answer
\[ \frac{1}{2}\left[3^n + 8n - 1\right] \]
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