Question

if the sum of the first 2n terms of the AP series 2,5 ,8......... is equal to the sum of the first n terms of the AP series 57 ,59,61.......then n is equal to ​

Answer 1

Step-by-step explanation:

We are given that

Two A.P series $\boxed{\mathsf{ 2,5 ,8.........}}$

let sum be $\bf{\mathsf{ {S}_{2n} }}$

$\boxed{\mathsf{57 ,59,61.......}}$

let its sum be as $\bf{\mathsf{ S_n }}$

According to Question

$\implies{\mathsf{S_{2n} = S_n }}$

Using Sum formula ,

$\boxed{\implies{\mathsf{ S_n = \frac{n}{2} \times (2n - 1)\times D}}}$

$\implies{\mathsf{\frac{2n}{2}[2 \times 2 + (2n -1) \times 3 ] = \frac{n}{2}[2 \times 57 + (n -1) \times 2 }}$

$\implies{\mathsf{2\times [4 + 6n - 3] = (114 +2n - 2)}}$

$\implies{\mathsf{ (4 + 6n -3) = 1/2 \times (114 +2n - 2) }}$

$\implies{\mathsf{ 6n + 1 = 57 + n - 1 }}$

$\implies{\mathsf{ 6n - n = 57 - 1}}$

$\implies{\mathsf{ 5n = 55 }}$

$\boxed{\huge{\mathsf{ n = 11 }}}$

$\rule{200}{2}$

$\implies{\underline{\mathcal{What \:is \:A.P \:? }}}$

A.P stands for Airthematic progression

Its an type of sequence in which the every next terms follows a specific pattern that is common difference