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

> [!TLDR] Equating the given expression to the arithmetic mean $\frac{a+b}{2}$ and simplifying isolates $a^{n-1}=b^{n-1}$, which forces $n=1$. > [!FORMULA] The arithmetic mean of two numbers $a$ and $b$ is: > $$\text{A.M.} = \frac{a+b}{2}$$ > We are given that $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}$ equals this A.M., and must find $n$. 1. Set up the equation: $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}} = \dfrac{a+b}{2}$. 2. Cross-multiply: $2(a^n+b^n) = (a+b)(a^{n-1}+b^{n-1})$. 3. Expand the right side: $(a+b)(a^{n-1}+b^{n-1}) = a^n + ab^{n-1} + a^{n-1}b + b^n$. 4. So the equation becomes: $2a^n+2b^n = a^n+b^n+ab^{n-1}+a^{n-1}b$. 5. Subtract $a^n+b^n$ from both sides: $a^n+b^n = ab^{n-1}+a^{n-1}b$. 6. Rearrange: $a^n - a^{n-1}b = ab^{n-1} - b^n$, i.e. $a^{n-1}(a-b) = b^{n-1}(a-b)$. …

If dfracaⁿ+bⁿaⁿ⁻¹+bⁿ⁻¹ is the arithmetic mean between a and b, then find the value of n.

Worked Examples · Example 15

\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}

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Equating the given expression to the arithmetic mean $\frac{a+b}{2}$ and simplifying isolates $a^{n-1}=b^{n-1}$ , which forces $n=1$ .

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Set up the equation: $\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}} = \dfrac{a+b}{2}$ .
Cross-multiply: $2(a^n+b^n) = (a+b)(a^{n-1}+b^{n-1})$ .
Expand the right side: $(a+b)(a^{n-1}+b^{n-1}) = a^n + ab^{n-1} + a^{n-1}b + b^n$ .
So the equation becomes: $2a^n+2b^n = a^n+b^n+ab^{n-1}+a^{n-1}b$ .
Subtract $a^n+b^n$ from both sides: $a^n+b^n = ab^{n-1}+a^{n-1}b$ .
Rearrange: $a^n - a^{n-1}b = ab^{n-1} - b^n$ , i.e. $a^{n-1}(a-b) = b^{n-1}(a-b)$ . …

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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 find the value of $n$ .

Concept understanding — Arithmetic Mean

What is the Arithmetic Mean?

The Precise Definition

Why It Works

A Quick Example …

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 find the value of nnn.

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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 find the value of $n$ .

Unlock everything free for 14 days