. find the sum of the …

Given: S_n = 3 + 7 + 13 + 21 + 31 + ...... + a_n-1 + a_n……….(i)

Also S_n = 3 + 7 + 13 + 21 + 31 + ...... + a_n-2 + a_n-1 + a_n  ……….(ii)

Subtracting eq. (i) from eq. (ii), 0 = 3 + ( 4 + 6 + 8 + 10 + ....... up to (n - 1) terms) - a_n 

{tex}\Rightarrow a _ { n } = 3 + \frac { n - 1 } { 2 } [ 2 \times 4 + ( n - 2 ) \times 2 ]{/tex}  

{tex}\Rightarrow a _ { n } = 3 + \frac { n - 1 } { 2 } [ 8 + 2 n - 4 ]{/tex}

{tex}\Rightarrow{/tex} a_n = 3 + (n - 1) (n + 2)

{tex}\Rightarrow{/tex} an = 3 + n² + n - 2

{tex}\Rightarrow{/tex} a_n = n² + n + 1

{tex}\therefore{/tex} {tex}{S_n} = \sum\limits_{k = 1}^n {{a_{_k}}} = \sum\limits_{k = 1}^n {({k^{^2}}} + k + 1){/tex}

= (1² + 1 + 1) + (2² + 2 + 1) + (3² + 3 + 1) + ...... +(n² + n + 1) 

= (1² + 2² + 3² + ....... + n²) + (1 + 2 + 3 + ...... + n) + n 

{tex}= \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + \frac { n ( n + 1 ) } { 2 } + n{/tex}

{tex}= n \left[ \frac { 2 n ^ { 2 } + 3 n + 1 + 3 n + 3 + 6 } { 6 } \right]{/tex}

{tex}= n \left[ \frac { 2 n ^ { 2 } + 6 n + 10 } { 6 } \right]{/tex}

{tex}= \frac { n } { 3 } \left( n ^ { 2 } + 3 n + 5 \right){/tex}