Question

If in the expansion of (x²-1/4)ⁿ. the coefficient of third term is 31, then the value of n is​

Answer 1

SOLUTION

GIVEN

In the expansion of

$\displaystyle \sf{ {\bigg( {x}^{2} - \frac{1}{4} \bigg)}^{n} }$

the coefficient of third term is 31

TO DETERMINE

The value of n

EVALUATION

Here the given binomial expansion is

$\displaystyle \sf{ {\bigg( {x}^{2} - \frac{1}{4} \bigg)}^{n} }$

The third term

$\displaystyle \sf{ t_3= {}^{n}C_2 \: { \big( {x}^{2} \big)}^{n - 2} {\bigg( - \frac{1}{4} \bigg)}^{2} }$

$\implies \displaystyle \sf{ t_3= {}^{n}C_2 \: . \: {x}^{2n - 4} \: . \: \frac{1}{16} }$

Therefore the coefficient of third term

$\displaystyle \sf{= {}^{n}C_2 \: .\: \frac{1}{16} }$

So by the given condition

$\displaystyle \sf{ {}^{n}C_2 \: .\: \frac{1}{16} = 31}$

$\implies\displaystyle \sf{ {}^{n}C_2 = 31 \times 16}$

$\implies\displaystyle \sf{ {}^{n}C_2 = {}^{32}C_2 }$

Comparing both sides we get n = 32

FINAL ANSWER

The required value of n = 32

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