Question

If n is an odd number then find the sum of the first n positions of the following range

1² + 2.2² + 3² + 2.4² + 5² + 2.6² + ...

Answer 1

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$\bold{\underline{Question-}}$


If n is an odd number then find the sum of the first n positions of the following range

1² + 2.2² + 3² + 2.4² + 5² + 2.6² + ...





$\bold{\underline{Answer-}}$


°•° n is an odd number

let n = 2p + 1 where, p is an Integer

Now, Sum of n terms in the category

= Sum of posts (2p +1)

= sum of posts {(p + 1) + p}

[1² + 3² + 5² + ...(p + 1) post ] + 2[2² + 4² + 6² + ...p. post]


$= \frac{1}{3} (p + 1)[4(p + 1) {}^{2} - 1] + 2 \frac{2}{3} .p(p + 1)(2p + 1)$


$= \frac{1}{3} (p + 1).[4(p + 1) {}^{2} - 1) +4p(2p + 1)]$


$= \frac{1}{3} (p + 1).(4 {p}^{2} + 8p + 4 - 1 + {8p}^{2} + 4p) \\ \\ = \frac{1}{3} (p + 1)( {12p}^{2} + 12p + 3) \\ \\ = (p + 1)( {4p}^{2} + 4p + 1) \\ \\ = (p + 1)(2p + 1) {}^{2}$

°•° n = 2p + 1 , •°• $p = \frac{n - 1}{2}$

$[ \frac{n - 1}{2} + 1] {n}^{2} \\ \\ \\ = \frac{1}{2} (n + 1) {n}^{2}$

Answer 2

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Kindly refer to given attachment !

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