The coefficients of $(r-1)$th, $r$th and $(r+1)$th terms in the expansion of $(x+1)^n$ are in the ratio $1 : 3 : 5$. Find $n$ and $r$.

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> [!TLDR]
> Writing the (r−1)th, rth, (r+1)th term coefficients as $\binom{n}{r-2},\binom{n}{r-1},\binom{n}{r}$ and using the consecutive-ratio identity twice gives $n=7$, $r=3$.
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> In the expansion of $(x+1)^n$, the general term is $T_{k+1}=\binom{n}{k}x^k$. So:
> - $(r-1)$th term $=T_{r-1}$, coefficient $=\binom{n}{r-2}$
> - $r$th term $=T_r$, coefficient $=\binom{n}{r-1}$
> - $(r+1)$th term $=T_{r+1}$, coefficient $=\binom{n}{r}$
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> Given $\binom{n}{r-2} : \binom{n}{r-1} : \binom{n}{r} = 1:3:5$.
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> **First ratio:** $\dfrac{\binom{n}{r-1}}{\binom{n}{r-2}} = 3$. Using $\dfrac{\binom{n}{k}}{\binom{n}{k-1}}=\dfrac{n-k+1}{k}$ with $k=r-1$:
> $$\frac{n-r+2}{r-1} = 3 \;\Rightarrow\; n-r+2 = 3r-3 \;\Rightarrow\; n = 4r-5 \quad 	ext{...(i)}$$
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> **Second ratio:** $\dfrac{\binom{n}{r}}{\binom{n}{r-1}} = \dfrac{5}{3}$. With $k=r$: …

Power	Expansion
$(x + y)^0$	$1$
$(x + y)^1$	$x + y$
$(x + y)^2$	$x^2 + 2xy + y^2$
$(x + y)^3$	$x^3 + 3x^2y + 3xy^2 + y^3$
$(x + y)^4$	$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

$k$	$\binom{4}{k}$	$x^{4-k}$	$y^k$	Term
0	$\binom{4}{0}=1$	$x^4$	$y^0$	$1 \cdot x^4$
1	$\binom{4}{1}=4$	$x^3$	$y^1$	$4x^3y$
2	$\binom{4}{2}=6$	$x^2$	$y^2$	$6x^2y^2$
3	$\binom{4}{3}=4$	$x^1$	$y^3$	$4xy^3$
4	$\binom{4}{4}=1$	$x^0$	$y^4$	$1 \cdot y^4$

Q.The coefficients of $(r-1)$ th, $r$ th and $(r+1)$ th terms in the expansion of $(x+1)^n$ are in the ratio $1 : 3 : 5$ . Find $n$ and $r$ .

Concept understanding — Binomial Theorem Expansion

The Problem That Started It All

The Pattern You Already Know

The Precise Statement

Where Do Those Coefficients Come From?

The General Term

What If It's Not Just $x$ and $y$ ? …

Q.The coefficients of (r−1)(r-1)(r−1)th, rrrth and (r+1)(r+1)(r+1)th terms in the expansion of (x+1)n(x+1)^n(x+1)n are in the ratio 1:3:51 : 3 : 51:3:5. Find nnn and rrr.

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Q.The coefficients of $(r-1)$ th, $r$ th and $(r+1)$ th terms in the expansion of $(x+1)^n$ are in the ratio $1 : 3 : 5$ . Find $n$ and $r$ .

Unlock everything free for 14 days