In an A.P., show that a_ m+n +a_ m-n =2a_m.

Question

In an A.P., show that $\text{a}_{\text{m}+\text{n}}+\text{a}_{\text{m}-\text{n}}=2\text{a}_\text{m}.$

✓

Answer

It is given that the sequence $<\text{a}_\text{n}>$ is an A.P. $\therefore\text{a}_\text{n}=\text{a}+(\text{n}-1)\text{d}\ .....(1)$ Similarly from (1) $\text{a}_{\text{m+n}}=\text{a}+(\text{m}+\text{n}-1)\text{d}\ .....(2)$ $\text{a}_{\text{m+n}}=\text{a}+(\text{m}-\text{n}-1)\text{d}\ .....(3)$ Adding (2) and (3) $\text{a}_{\text{m+n}}+\text{a}_{\text{m}-\text{n}}=(\text{a}+(\text{m+n}-1)\text{d})(\text{a}+(\text{m}-\text{n}-1)\text{d})$ $=2\text{a}+(\text{m+n}-1+\text{m}-\text{n}-1)\text{d}$ $=\text{2a}+\text{2d}(\text{m}-1)$ $=2(\text{a}+(\text{m}-1)\text{d})$ $=\text{2a}_\text{m}$ Hence proved.

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 Arithmetic Progressions→Arithmetic Progressions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve the following equation: $4\sin^{2}\text{x}-8\cos\text{x}+1=0$

→

If $\Big(\frac{\text{a}}{3}+1,\text{b}-\frac{2}{3}\Big)=\Big(\frac{5}{3},\frac{1}{3}\Big),$ find the values of a and b.
f(x + 1, 1) = (3, y - 2), find the values of x and y.

→

Find the equation of the set of the point P, the sum of whose distance from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to 10.

→

Differentiate the following functions with respect to x:$\frac{\text{x}^2-\text{x}+1}{\text{x}^2+\text{x}+1}$

→

The ratio of the sum of $m$ and $n$ terms of an A.P. is $m^2: n^2$ Show that the ratio of $m^{\text {th }}$ and $n^{\text {th }}$ term is $(2 m-1):(2 n-1)$.

→

In a cricket team tournament 16 teams participated. A sum of ₹ 8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹ 275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?

→

Two plants A and B of a factory show following results about the number of workers and the wages paid to them: