Question

if S1,S2,S3 be the sum of n terms of three A.P ,then the first term of each being 1 and respectI've common difference 1,2,3 prove that S1+S2=2S2

Answer 1

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

Given three AP series of n terms having first term 1 and common difference 1, 2 and 3 respectively.

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

Case :- 1

First term, a = 1

Common difference, d = 1

Number of terms = n

Sum of n terms is

$\rm :\longmapsto\:S_1\:=\dfrac{n}{2} \bigg(2 \:(1)\:+\:(n\:-\:1)\:(1) \bigg)$

$\rm :\longmapsto\:S_1\:=\dfrac{n}{2} \bigg(2 \:+\:(n\:-\:1) \bigg)$

$\rm :\longmapsto\:S_1\:=\dfrac{n}{2} \bigg(2 \:+\:n\:-\:1 \bigg)$

$\bf :\longmapsto\:S_1\:=\dfrac{n}{2} \bigg(1 \:+\:n\: \bigg) - - - (1)$

Case :- 2

First term, a = 1

Common difference, d = 2

Number of terms = n

Sum of n terms is

$\rm :\longmapsto\:S_2\:=\dfrac{n}{2} \bigg(2 \:(1)\:+\:(n\:-\:1)\:(2) \bigg)$

$\rm :\longmapsto\:S_2\:=\dfrac{n}{2} \bigg(2 \:+\:2n\:-\:2\: \bigg)$

$\rm :\longmapsto\:S_2\:=\dfrac{n}{2} \bigg(\:2n\: \bigg)$

$\bf :\longmapsto\:S_2\:= {n}^{2} - - - - (2)$

Case :- 3

First term, a = 1

Common difference, d = 3

Number of terms = n

Sum of n terms is

$\rm :\longmapsto\:S_3\:=\dfrac{n}{2} \bigg(2 \:(1)\:+\:(n\:-\:1)\:(3) \bigg)$

$\rm :\longmapsto\:S_3\:=\dfrac{n}{2} \bigg(2 \:+\:3n\:-\:3\: \bigg)$

$\bf :\longmapsto\:S_3\:=\dfrac{n}{2} \bigg(\:3n\:-\:1\: \bigg) - - - - (3)$

Now, Consider

$\rm :\longmapsto\:S_1 + S_3$

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

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

$\rm \: = \: \: \dfrac{n}{2} \bigg(4n \: \bigg)$

$\rm \: = \: \: {2n}^{2}$

$\rm \: = \: \: 2S_2$

Hence, Proved

Additional Information :-

↝ 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.