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If the coefficients of 5th , 6th and 7th terms in the expansio

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If the coefficients of 5th , 6th and 7th terms in the expansion of $(1 - x)^{n}$ are in A.P then find the value of n.

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Emma Discussed

If the coefficients of 5th , 6th and 7th terms in the expansion of

(1 - x)^{n}

are in A.P then find the value of n.

15 mins ago

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Text solutionVerified

It can be

(1 + x)^{n}

only as

(1 - x)^{n}

terms are

+

ve/

-

ve alternatively hence can't be in A.P.,

General term,

T_{rH} =^{n} C_{r} \cdot x^{r}

\Rightarrow T_{5} =^{n} C_{4} \cdot x^{4}, T_{6} =^{n} C_{5} \cdot x^{5}, T_{7} =^{n} C_{6} \cdot x^{6}

Coefficients are in A.P.

\Rightarrow 2^{n} C_{5} =^{n} C_{4} +^{n} C_{6}

\Rightarrow \frac{2 n !}{5 ! ( n - 5 )!} = \frac{n !}{4 ! ( n - 4 )!} + \frac{n !}{6 ! ( n - 6 )!}

\Rightarrow \frac{2}{5 ( n - 5 )} = \frac{1}{( n - 4 ) ( n - 5 )} + \frac{1}{6 \times 5}

\Rightarrow 2 (n - 4) = 5 + \frac{1}{6} (n - 4) (n - 5)

\Rightarrow 12 (n - 4) = 30 + n^{2} - 9 n + 20

12 n - 48 = n^{2} - 9 n + 50

\Rightarrow n^{n} - 21 n + 98 = 0

\Rightarrow n^{2} - 14 n - 7 n + 98 = 0

\Rightarrow n (n - 14) - 7 (n - 14) = 0

\Rightarrow (n - 7) (n - 14) = 0

n = 7

n = 14

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Question Text	If the coefficients of 5th , 6th and 7th terms in the expansion of $(1 - x)^{n}$ are in A.P then find the value of n.
Answer Type	Text solution:1
Upvotes	150