Question

Solve this question please.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given series is

$\rm :\longmapsto\:S_n = 1.3 + 3.5 + 5.7 + - - - -$

Since, this series is sum of multiplication of corresponding terms of two series

$\rm :\longmapsto\:1,3,5, - - - -$

and

$\rm :\longmapsto\:3,5,7, - - - -$

As

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

So,

↝ nᵗʰ term of the given series is,

$\rm :\longmapsto\:a_n = {n}^{th}term \: of \: (1,3,5, - - ) \times (3,5,7, - - - )$

$\rm :\longmapsto\:a_n = \bigg(1 + (n - 1)2 \bigg) \bigg(3 + (n - 1)2 \bigg)$

$\rm :\longmapsto\:a_n = \bigg(1 + 2n - 2 \bigg) \bigg(3 + 2n - 2 \bigg)$

$\rm :\longmapsto\:a_n = \bigg( 2n - 1 \bigg) \bigg(2n + 1 \bigg)$

$\bf\implies \:a_n = {4n}^{2} - 1$

So, Sum of n terms of given series is

$\red{\rm :\longmapsto\:S_n = \displaystyle\sum_{n=1}^n a_n \: }$

$\red{\rm :\longmapsto\:S_n = \displaystyle\sum_{n=1}^n ( {4n}^{2} - 1) \: }$

$\red{\rm :\longmapsto\:S_n =4 \displaystyle\sum_{n=1}^n {n}^{2} - \displaystyle\sum_{n=1}^n 1 \: }$

We know,

$\green{\boxed{ \tt{ \: \displaystyle\sum_{n=1}^n {n}^{2} = \frac{n(n + 1)(2n + 1)}{6} \: }}}$

and

$\green{\boxed{ \tt{ \: \displaystyle\sum_{n=1}^n 1 = n}}}$

So, on substituting the values, we get

$\red{\rm :\longmapsto\:S_n = 4 \times \dfrac{n(n + 1)(2n + 1)}{6} - n \: }$

$\red{\rm :\longmapsto\:S_n = 2 \times \dfrac{n(n + 1)(2n + 1)}{3} - n \: }$

$\red{\rm :\longmapsto\:S_n = \dfrac{2n(n + 1)(2n + 1)}{3} - n \: }$

So,

$\boxed{ \tt{ \: 1.3 + 3.5 + 5.7 + - - n \: terms = \frac{2n(n + 1)(2n + 1)}{3} - n \: }}$

Hence,

• Option (d) is correct

Additional Information :-

$\green{\boxed{ \tt{ \: \displaystyle\sum_{n=1}^n {n} = \frac{n(n + 1)}{2} \: }}}$

$\green{\boxed{ \tt{ \: \displaystyle\sum_{n=1}^n {n}^{3} = {\bigg[\dfrac{n(n + 1)}{2} \bigg]}^{2} \: }}}$