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In the expansion of the expression { ( x+a ) }^{ 15 }, if the

Question

In the expansion of the expression $(x + a)^{15}$ , if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest?

T_{6}

T_{7}

T_{8}

T_{9}

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Charlotte Discussed

In the expansion of the expression

(x + a)^{15}

, if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest?

5 mins ago

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5 mins ago

Text solutionVerified

From the given condition, one get

(_{15} C_{10} x^{2} a^{10})^{2} = (_{15} C_{7} x^{8} a^{7}) (_{15} C_{11} x^{4} a^{11}) \Rightarrow \frac{a}{x} = \frac{75}{77}

Now

(x + a)^{15} = x^{15} (1 + \frac{a}{x})^{15}

consider the expansion of

(1 + \frac{a}{x})^{15}

T_{r} = T_{r + 1} \Rightarrow \frac{r}{( n - r + 1 ) ( \frac{a}{x} )} = 1

\Rightarrow r = 16 \frac{75}{77} - r \frac{75}{77} \Rightarrow r = \frac{16 \frac{75}{77}}{1 + \frac{75}{77}} = 7

r = 7 \Rightarrow T_{7} < T_{8}

and

r = 8 \Rightarrow T_{8} > T_{9}

Hence

T_{8}

is the greater term

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Question Text	In the expansion of the expression $(x + a)^{15}$ , if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest?
Answer Type	Text solution:1
Upvotes	150