The $n$th term of the series $1+4+13+40+121+364+.......$ is
- $\cfrac { 1 }{ 2 } \left( { 3 }^{ n }+1 \right) $
- $\cfrac { 1 }{ 2 } \left( { 3 }^{ n }-1 \right) $
- $\left( \cfrac { { 2 }^{ n }+1 }{ 2 } \right) 2$
- ${ 3 }^{ n }-1$
Reveal answer
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B
Correct answer
Explanation
The series is 1, 4, 13, 40, 121, 364. This is a geometric series sum: 1 + 3 + 9 + 27 + 81 + 243. The nth term is the sum of a geometric progression with first term 1 and ratio 3: (3^n - 1) / (3 - 1) = (3^n - 1) / 2.
AI explanation
By examining the series, each term can be rewritten by adding 0.5 to it, transforming the sequence into 1.5, 4.5, 13.5, 40.5, and so on. This altered sequence is a geometric progression with a first term of 1.5 and a common ratio of 3, meaning its nth term is 1.5 * 3^(n-1), which equals 3^n / 2. Subtracting the 0.5 we initially added gives the formula for the nth term of the original series as (3^n - 1) / 2. The correct nth term is 1/2(3^n - 1).