The $n$th term of the series $5+7+13+31+85+\cdots$ is
- ${ 3 }^{ n-1 }+2$
- $3+{ 3 }^{ n-1 }$
- $4+{ 3 }^{ n-1 }$
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None of these
Reveal answer
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C
Correct answer
Explanation
The series is 5, 7, 13, 31, 85. The differences are 2, 6, 18, 54. These are 2 * 3^0, 2 * 3^1, 2 * 3^2, 2 * 3^3. The nth term is 5 + sum(2 * 3^(k-1)) for k=1 to n-1. This equals 5 + 2 * (3^(n-1) - 1) / (3 - 1) = 5 + 3^(n-1) - 1 = 4 + 3^(n-1).
AI explanation
Test the given formula 4 + 3^(n-1) by substituting the term numbers into n. For n = 1, the formula gives 4 + 3^0 = 5; for n = 2, it gives 4 + 3^1 = 7; for n = 3, it gives 4 + 3^2 = 13. This matches the given sequence 5 + 7 + 13 + ..., so the nth term is 4 + 3^(n-1).