Question

If m times $m^{th}$ term of an A.P. is equal to n times its $n^{th}$ term, show that the (m + n) term of the A.P. is zero. [3 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks

Let a be the first term and d be the common difference of the given A.P.

$m\times a_m=n\times a_n$

$m(a+m(m–1\left)d\right)=n(a+n(n–1\left)d\right)$

$m(a+m(m–1\left)d\right)–n(a+n(n–1\left)d\right)=0$

$a(m–n)+\left[m(m–1)–n(n–1)\right]d=0$

$a(m–n)+\left[\right(m–n\left)\right(m+n–1\left)\right]d=0$

Taking (m-n ) common

$(m–n)\left[a+(m+n–1)d\right]=0$

$a+\left(\right(m+n)–1)d=0$

$a_{m+n}=0$