Question

Find the sum 5 x 7 + 9 × 11 + 11 × 13 + ---
upto n terms.​

Answer 1

sum of given expression is n(n + 1)(4n + 26)/3 + 15n

we have to find the sum of 5 × 7 + 7 × 9 + 9 × 11 + 11 × 13 ..... upto n terms.

first find the Tn and then apply sigma.

Tn = (2n + 3)(2n + 5) , where n =1, 2, 3, ...

now,sum = $\Sigma{T_n}$

= $\Sigma{(2n+3)(2n+5)}$

= $\Sigma{4n^2+16n+15}$

= $4\Sigma{n^2}+16\Sigma{n}+15\Sigma$

= 4n(n + 1)(2n + 1)/6 + 16n(n + 1)/2 + 15n

= 2n(n + 1)(2n + 1)/3 + 8n(n + 1) + 15n

= n(n + 1)[2(2n + 1)/3 + 8] + 15n

= n(n + 1)[(4n + 2 + 24)/3 ] + 15n

= n(n + 1)(4n + 26)/3 + 15n

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Answer 2

Tn = (2n + 3)(2n + 5) , where n =1, 2, 3, ...

= 4n(n + 1)(2n + 1)/6 + 16n(n + 1)/2 + 15n

= 2n(n + 1)(2n + 1)/3 + 8n(n + 1) + 15n

= n(n + 1)[2(2n + 1)/3 + 8] + 15n

= n(n + 1)[(4n + 2 + 24)/3 ] + 15n

= n(n + 1)(4n + 26)/3 + 15n