First slide

Binomial theorem for positive integral Index

Question

The two consecutive terms in the expansion of ${(3 + 2 x)}^{74}$ , which have equal coefficients are

Moderate

A

$7^{th}$ and $8^{th}$
B

$11^{th}$ and $12^{th}$
C

$30^{th}$ and $31^{s t}$
D

None of these

Solution

Given expansion ${(3 + 2 x)}^{74}$ lets the consecutive terms are
$(∴ T_{r + 1} =^{n} C_{r} x^{n - r} {(a)}^{r}$ be the expansion of ${(x + a)}^{n})$

Here $T_{r + 1} =^{74} C_{r} 3^{74 - r} {(2 x)}^{r}$

And $T_{r} =^{74} C_{r - 1} 3^{75 - r} {(2 x)}^{r - 1}$

The coefficients of these two terms are equal according to given problem

If $^{74} C_{r} 3^{74 - r} 2^{r} =^{74} C_{r - 1} 3^{75 - r} 2^{r - 1}$

$\begin{matrix} \Rightarrow \frac{^{74} C_{r}}{^{74} C_{r - 1}} = \frac{3^{75 - r} 2^{r - 1}}{3^{74 - r} 2^{r}} \\ \Rightarrow \frac{^{74} C_{r}}{^{74} C_{r - 1}} = \frac{3}{2} (∴ \frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}) \\ \Rightarrow \frac{75 - r}{r} = \frac{3}{2} \\ ∴ r = 30 \end{matrix}$

So, $T_{31}$ and $T_{30}$ have equal coefficients.

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