Answer :

To prove: that. If the coefficients of x^{2} and x^{3} in the expansion of (3 + px)^{9} are the same then .

Formula Used:

General term, T_{r+1} of binomial expansionis given by,

T_{r+1} ^{n}C_{r} x^{n-r} y^{r} where

^{n}C_{r}

Now, finding the general term of the expression, (3 + px)^{9} , we get

T_{r+1} ^{9}C_{r}

For finding the term which has in it, is given by

r=2

Thus, the coefficients of x^{2} are given by,

T_{3} ^{9}C_{2}

T_{3} ^{9}C_{2}

For finding the term which has in it, is given by

r=3

Thus, the coefficients of x^{3} are given by,

T_{3} ^{9}C_{3}

T_{3} ^{9}C_{3}

As the coefficients of x^{2} and x^{3} in the expansion of (3 + px)^{9} are the same.

^{9}C_{3} ^{9}C_{2}

^{9}C_{3} ^{9}C_{2}

Thus, the value of p for which coefficients of x^{2} and x^{3} in the expansion of (3 + px)^{9} are the same is

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