Question

The pth term of an arithmetic progression is m and q th term is n respectively having first term a and common difference d, then value of common difference is______.
A) m/q+n/ p B)m-n/p-q C)m+n/p+q D)mn/pq

Answer 1

Answer:

B) m-n/p-q

Step-by-step explanation:

pth term is m

m= a+(p-1)d {say (i)}

n= a+(q-1)d {say (ii)}

subtracting (ii) from (i)

m= a+(p-1)d

n= a+(q-1)d

------------------

m-n = (p-q)d

m-n/p-q=d

Answer 2

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

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.

Given that,

↝ pᵗʰ term of an arithmetic progression is m

$\rm :\longmapsto\: a_{p} = m$

$\rm :\longmapsto\:a + (p - 1)d = m - - - (1)$

Also, given that

↝ qᵗʰ term of an arithmetic progression is n

$\rm :\longmapsto\: a_{q} = n$

$\rm :\longmapsto\:a + (q - 1)d = n - - - (2)$

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

$\rm :\longmapsto\:(p - 1)d - (q - 1)d = m - n$

$\rm :\longmapsto\:(p - 1 - q + 1)d = m - n$

$\rm :\longmapsto\:(p - q)d = m - n$

$\bf\implies \:d = \dfrac{m - n}{p - q}$

Hence,

• Option B is correct.

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Additional Information

↝ 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.