Question

For an A.P. Sp = q Sq = p , then Sp+q = ​

Answer 1

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

Given that,

The sum of first p terms of an AP is q and sum of first q terms is p.

To find the sum of first p + q terms of an AP.

Let assume that first term of an AP series is a and common difference is d.

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

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

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.

As it is given that,

$\rm \: S_p = q$

$\rm \: \dfrac{p}{2}\bigg(2a + (p - 1)d \bigg) = q$

$\rm\implies \:\rm \: 2ap + p(p - 1)d = 2q - - - (1)$

Further given that,

$\rm \: S_q = p$

$\rm \: \dfrac{q}{2}\bigg(2a + (q - 1)d \bigg) = p$

$\rm\implies \:\rm \: 2aq + q(q - 1)d = 2p - - - (2)$

On Subtracting equation (2) from equation (1), we get

$\rm \: 2a(p - q) + d\bigg(p(p - 1) - q(q - 1)\bigg) = 2q - 2p$

$\rm \: 2a(p - q) + d\bigg( {p}^{2} - p - {q}^{2} + q \bigg) = - 2(p - q)$

$\rm \: 2a(p - q) + d\bigg( {p}^{2}- {q}^{2} - p+ q \bigg) = - 2(p - q)$

$\rm \: 2a(p - q) + d\bigg( (p - q)(p + q) -( p - q ) \bigg) = - 2(p - q)$

$\rm \: (p - q)\bigg(2a + d(p + q - 1) \bigg) = - 2(p - q)$

$\rm\implies \:2a + (p + q - 1)d = - 2 - - - (3)$

Now, Consider

$\rm \: S_{p + q}$

$\rm \:=  \: \: \dfrac{p + q}{2}\bigg(2a + (p + q - 1)d \bigg)$

On substituting the value from equation (3), we get

$\rm \:=  \: \: \dfrac{p + q}{2}\bigg( - 2\bigg)$

$\rm \: =  \: - (p + q)$

Hence,

$\rm\implies \: \boxed{\sf{  \:\: \: \rm \: S_{p + q} \: = \: - \: (p \: + \: q) \: \: \: }}$

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Additional Information

↝ nᵗʰ term of an arithmetic sequence is,

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

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.

Answer 2

8×8=020

×8×5×8×=as

as=52/92

1029=ap

sp=1092 /8

239