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

$\begin{array}{1 1}(A)\;n=7,r=3\\(B)\;n=3,r=7\\(C)\;n=4,r=8\\(D)\;n=5,r=3\end{array} $

1 Answer

  • General term $T_{k+1}=C(n,k)x^{n-k}$
  • $C(n,r)=\large\frac{n!}{r!(n-r)!}$
General term in the expansion of $(x+1)^n$ is
Putting $T_{k+1}=T_{r-1}$
Coefficient of $T_{r-1}$ is $C(n,r-2)$-----(1)
Putting $T_{k+1}=T_r$
Coefficient of $T_r$ is $C(n,r-1)$-----(2)
Coefficient of $T_{r+1}$ is $C(n,r)$-----(3)
According to the problem
$C(n,r-2) : C(n,r-1) : C(n,r) =1 : 3 : 5$
If $\large\frac{C(n,r-2)}{1}=\frac{C(n,r-1)}{3}$
Or $3C(n,r-2)=C(n,r-1)$
Or $3\large\frac{n!}{(r-2)!(n+2-r)!}=\frac{1}{(r-1)(r-2)!(n+1-r)!}$
Again if If $\large\frac{C(n,r-1)}{3}=\frac{C(n,r)}{5}$
Or $5C(n,r-1)=3C(n,r)$
Or $5\large\frac{n!}{(r-1)!(n+1-r)!}=$$3\large\frac{n!}{r!(n-r)!}$
From (4) and (5)
From (5) $8r=21+3$
$\therefore n=7,r=3$
Hence (A) is the correct answer.
answered Jun 23, 2014 by sreemathi.v

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