Show that the sum of (m+n)^ th and (m-n)^ th terms of an A.P. is equal to twice the m^ th terms.

Here, a_ m+n =a+(m+n-1) d …..(i) and a_ m-n =a+(m-n-1) d …..(ii) To prove: a_ m+n +a_ m-n =2 a_m Adding eq. (i) and (ii), we get a_ m+n +a_ m-n =a+(m+n-1) d+a+(m-n-1) d a_ m+n +a_ m-n =a+m d+n d-d+a+m d-n d-d a_ m+n +a_ m-n =2 a+2 m d+-2 d a_ m+n +a_ m-n =2(a+m d-d) a_ m+n +a_ m-n =2[a+(m-1) d] a_ m+n +a_ m-n =2 a_m

Show that the sum of (m+n)^ th and (m-n)^ th terms of an A.P. is …

Download App

Question

Show that the sum of $(m+n)^{\text {th }}$ and $(m-n)^{\text {th }}$ terms of an A.P. is equal to twice the $m^{\text {th }}$ terms.

✓

Answer

Here, $a_{m+n}=a+(m+n-1) d$ …..(i) and $a_{m-n}=a+(m-n-1) d$ …..(ii)
To prove: $a_{m+n}+a_{m-n}=2 a_m$
Adding eq. (i) and (ii), we get
$a_{m+n}+a_{m-n}=a+(m+n-1) d+a+(m-n-1) d$
$\Rightarrow a_{m+n}+a_{m-n}=a+m d+n d-d+a+m d-n d-d$
$\Rightarrow a_{m+n}+a_{m-n}=2 a+2 m d+-2 d$
$\Rightarrow a_{m+n}+a_{m-n}=2(a+m d-d)$
$\Rightarrow a_{m+n}+a_{m-n}=2[a+(m-1) d]$
$\Rightarrow a_{m+n}+a_{m-n}=2 a_m$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 3 marks)" from SEQUENCES AND SERIES→SEQUENCES AND SERIES — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

SEQUENCES AND SERIES — MATHS STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions