The $n$th term of the series $1+2+5+12+25+\cdots $ is
- $\left( n-1 \right) \left( n-2 \right) $
- $\dfrac { 1 }{ 3 } n\left( n-1 \right) \left( n-2 \right) +n$
- $n$
-
None of these
The series is 1, 2, 5, 12, 25. Differences are 1, 3, 7, 13. Second differences are 2, 4, 6. This indicates a cubic sequence. Testing n=1, 2, 3 gives the correct values for the formula 1/3 * n(n-1)(n-2) + n.
Substitute n equals 1 into the given options to see which equals the first term of the series. For option B, (1 divided by 3) times 1 times (1 - 1) times (1 - 2) + 1 equals 1, matching the first term. Checking the second term with n equals 2 gives (1 divided by 3) times 2 times 1 times 0 + 2 equals 2, and the third term gives (1 divided by 3) times 3 times 2 times 1 + 3 equals 5, matching the series perfectly. Therefore, the nth term is (1 divided by 3) times n times (n - 1) times (n - 2) plus n.