Question

4 + 6 + 9 + 13 + 18 + ...

Answer 1

Let \[T_n\]  be the nth term and  \[S_n\] be the sum of n terms of the given series. 

Thus, we have:

\[S_n = 4 + 6 + 9 + 13 + 18 + . . . + T_{n - 1} + T_n\] ...(1)

Equation (1) can be rewritten as:

\[S_n = 4 + 6 + 9 + 13 + 18 + . . . + T_{n - 1} + T_n\]  ...(2)

On subtracting (2) from (1), we get:

\[S_n = 4 + 6 + 9 + 13 + 18 + . . . + T_{n - 1} + T_n \]

\[ S_n = 4 + 6 + 9 + 13 + 18 + . . . + T_{n - 1} + T_n \]

\[ 0 = 4 + \left[ 2 + 3 + 4 + 5 + 6 + . . . + \left( T_n - T_{n - 1} \right) \right] - T_n\]

The sequence of difference between successive terms is 2, 3, 4, 5,...
We observe that it is an AP with common difference 1 and first term 2.
Now,

\[4 + \left[ \frac{\left( n - 1 \right)}{2}\left\{ 4 + \left( n - 2 \right)1 \right\} \right] - T_n = 0\]

\[ \Rightarrow 4 + \left[ \frac{\left( n - 1 \right)}{2}\left( n + 2 \right) \right] - T_n = 0\]

\[ \Rightarrow 4 + \left[ \frac{n^2 + n}{2} - 1 \right] - T_n = 0\]

\[ \Rightarrow \left[ \frac{n^2}{2} + \frac{n}{2} + 3 \right] = T_n\]

\[\because S_n = \sum^n_{k = 1} T_k \]

\[ \therefore S_n = \sum^n_{k = 1} \left( \frac{k^2}{2} + \frac{k}{2} + 3 \right)\]

\[ = \frac{1}{2} \sum^n_{k = 1} k^2 + \frac{1}{2} \sum^n_{k = 1} k + \sum^n_{k = 1} 3\]

\[ = \frac{n\left( n + 1 \right)\left( 2n + 1 \right)}{2 \times 6} + \frac{n\left( n + 1 \right)}{2 \times 2} + 3n\]

\[ = n\left( \frac{2 n^2 + 3n + 1 + 3n + 3 + 36}{12} \right)\]

\[ = \frac{n}{12}\left( 2 n^2 + 6n + 40 \right)\]

\[ = \frac{n}{6}\left( n^2 + 3n + 20 \right)\]