If $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the arithmetic mean between $a$ and $b$, then the value of $n$ is :

(a) 1
(b) $-1$
(c) $\dfrac{1}{2}$
(d) $-\dfrac{1}{2}$

Question

If $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the arithmetic mean between $a$ and $b$, then the value of $n$ is :

(a) 1
(b) $-1$
(c) $\dfrac{1}{2}$
(d) $-\dfrac{1}{2}$

Accepted Answer

>

> [!TLDR]
> Substituting small integer values of $n$ shows the expression equals the AM of $a,b$ exactly when $n=1$.
>
> The arithmetic mean of $a$ and $b$ is $\dfrac{a+b}{2}$. We need
> $$\dfrac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}}=\dfrac{a+b}{2}.$$
>
> Try $n=1$: the left side becomes
> $$\dfrac{a^{1}+b^{1}}{a^{0}+b^{0}}=\dfrac{a+b}{1+1}=\dfrac{a+b}{2},$$ …

What you need	Formula
nth term	$T_n = a + (n-1)d$
Sum of n terms	$S_n = \frac{n}{2}[2a + (n-1)d]$
Sum using last term	$S_n = \frac{n}{2}(a + l)$
Common difference	$d = T_{n+1} - T_n$

Q.If $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the arithmetic mean between $a$ and $b$ , then the value of $n$ is :

Concept understanding — Arithmetic Progression

Arithmetic Progression: The Pattern of Equal Steps

The Core Idea

The General Term (nth term)

Why "Arithmetic"?

Sum of the First n Terms

Quick Reference

Q.If an+bnan−1+bn−1\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}an−1+bn−1an+bn​ is the arithmetic mean between aaa and bbb, then the value of nnn is :

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Q.If $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the arithmetic mean between $a$ and $b$ , then the value of $n$ is :

Unlock everything free for 14 days