Question

If m times the m th term of an A.P is equal to n times it's n th term then find the(m+n)th term

Answer 1

Solution:

Let the first term of AP = a
and common difference = d

or, mth term = a + (m-1)d
or, nth term = a + (n-1) d

m times m th term of an AP = n times n th term of an AP.

Given that:
or, m{a +(m-1)d} = n{a + (n -1)d}
or, am + m²d -md = an + n²d - nd
or, am - an + m²d - n²d -md + nd = 0
or, a(m-n) + (m²-n²)d - (m-n)d = 0
or, a(m-n) + {(m-n)(m+n)}d -(m-n)d = 0
or, a(m-n) + {(m-n)(m+n) - (m-n)} d = 0
or, a(m-n) + (m-n)(m+n -1) d = 0 [Taking (m-n) common]
or, (m-n){a + (m+n-1)d} = 0 [Taking (m-n) common]
or, a + (m+n -1)d = 0/(m-n)
or, a + (m+n -1)d = 0

(Answer)