Question

The nth term of an arithmetic progression is equal to 2m and the mth term of that arithmetic progression is 2n. What
is the (n+ m) th term?
​

Answer 1

$\textbf{Given:}$

$\text{In an A.P t_n=2m \;\text{and}\; t_m=2n }$

$\textbf{To find:}$

$t_{m+n}$

$\textbf{Solution:}$

$\text{Formula used:}$

$\text{The n th term of the A.P a, a+d, a+2d, ....... is}$

$\boxed{\bf\,t_n=a+(n-1)d}$

$t_n=2m\implies\,a+(n-1)d=2m$...(1)

$t_m=2n\implies\,a+(m-1)d=2n$...(2)

$(1)-(2)\implies$

$(n-1)d-(m-1)d=2m-2n$

$nd-d-md+d=2m-2n$

$nd-md=2m-2n$

$(n-m)d=-2(n-m)$

$\implies\bf\,d=-2$

$\text{Put d=-2 in (1), we get}$

$a+(n-1)d=2m$

$a+(n-1)(-2)=2m$

$a-2n+2=2m$

$\implies\,a=2m+2n-2$

$\text{Now,}$

$t_{m+n}$

$=a+(m+n-1)d$

$=2m+2n-2+(m+n-1)(-2)$

$=2m+2n-2-2m-2n+2$

$=0$

$\therefore\textbf{m+n th term is 0}$

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