if S1,S2,S3 be the sum of n terms of three A.P ,then the first term of

1.	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» $\large\underline{\sf{Solution-}}$ 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.

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»

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

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,

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