Find the numerically greatest term in the expansion $(3 - 5 x)^{15}$ when $x = \frac{1}{5}$

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Solution

Greatest digit to be find $(3 - 5 x)^{15}$ when $x = \frac{1}{5}$
$T_{x} =^{15} C_{r} (3)^{15 - r} (- 5 x)^{r}$
$T_{x} =^{15} C_{r} (3)^{15 - r} (- 1)^{r}$
$x = + \frac{1}{5}$ and $(- 1)^{r}$ will chnage sign.
$T_{x} =^{15} C_{r} 3^{15 - r}$
Maximum is obtained when both are maximum
Therefore, $r = 7$
Maximum term= $^{15} C_{r} (3)^{15 - r}$
when r=7
$T_{x} =^{15} C_{7} (3)^{15 - 7}$
$T_{x} =^{15} C_{7} (3)^{8}$
$T_{x} = 42, 220, 035$
$T_{x} = 4.2 \times 10^{7}$

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