Question

For a A.P, if (m+n) th term is p and (m−n) th term is q, then m th term is

Answer 1

Step-by-step explanation:

let 1st term = a

and common difference = d

(m+n) th term is p

=> a + (m+n-1)d = p

=> a = p - (m+n-1)d ----(1)

(m-n) th term is p

=> a + (m-n-1)d = q

=> p - (m+n-1)d + (m-n-1)d = q [putting a from (1)]

=> (-m-n+1+m-n-1)d = q-p

=> -2nd = q-p

=> d = (p-q)/2n

putting d in (1),

a = p - (m+n-1)d = p - (m+n-1)(p-q)/2n

mth term

= a + (m-1)d

= p - (m+n-1)(p-q)/2n + (m-1)(p-q)/2n

= p + (-m-n+1+m-1)(p-q)/2n

= p - n(p-q)/2n

= p - (p-q)/2

= (2p - p + q)/2

= (p+q)/2