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Binomial Theorem

If the coeffi...

Question

If the coefficients of $r^{t h}$ and $(r + 1)^{t h}$ terms in the expansion of $(3 + 7 x)^{29}$ are equal, then the value of $r$ is

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Solution

Given : $(3 + 7 x)^{29}$
General term is
$T_{r + 1} =^{29} C_{r} (3)^{29 - r} (7 x)^{r}$
Coefficient of $(r + 1)^{t h}$ term
$=^{29} C_{r} \times 3^{29 - r} \times (7)^{r}$
Coefficient of $r^{t h}$ term
$=^{29} C_{r - 1} \times 3^{30 - r} \times (7)^{r - 1}$

According to question,
$^{29} C_{r} \times 3^{29 - r} \times (7)^{r} =^{29} C_{r - 1} \times 3^{30 - r} \times (7)^{r - 1} \Rightarrow \frac{^{29} C_{r}}{^{29} C_{r - 1}} = \frac{3}{7} \Rightarrow \frac{30 - r}{r} = \frac{3}{7} ∴ r = 21$

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Binomial Theorem

Standard XI Mathematics

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