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If the coefficients of 2^{nd}, 3^{rd} and 4^{th} terms in the

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If the coefficients of $2^{n d}$ , $3^{r d}$ and $4^{t h}$ terms in the expansion of $(1 + x)^{2 n}$ are in A.P., show that $2 n^{2} - 9 n + 7 = 0$ .

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If the coefficients of

2^{n d}

3^{r d}

and

4^{t h}

terms in the expansion of

(1 + x)^{2 n}

are in A.P., show that

2 n^{2} - 9 n + 7 = 0

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Text solutionVerified

Given the expression

(1 + x)^{2 n}

According to the condition

^{2 n} C_{1},^{2 n} C_{2},^{2 n} C_{3}

are in

A . P

so,

2 \times^{2 n} C_{2} =^{2 n} C_{1} +^{2 n} C_{3}

using

^{n} C_{r} = \frac{n !}{r ! ( n - r )!}

solving,

= 2 \times \frac{( 2 n )!}{2 ! ( 2 N - 2 )!} = \frac{( 2 n )!}{1 ! ( 2 n - 1 )!} + \frac{( 2 n )!}{3 ! ( 2 n - 3 )!}

= 2 \times \frac{2 n ( 2 n - 1 )}{2} = \frac{2 n}{1} + \frac{2 n ( 2 n - 1 ) ( 2 n - 2 )}{1 \times 2 \times 3}

= 2 n - 1 = 1 + \frac{( 2 n - 1 ) ( n - 1 )}{3}

= 6 n - 3 = 3 n + 2 n^{2} - 3 n + 1

= 2 n^{2} - 9 n + 7 = 0

Hence, proved

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Question Text	If the coefficients of $2^{n d}$ , $3^{r d}$ and $4^{t h}$ terms in the expansion of $(1 + x)^{2 n}$ are in A.P., show that $2 n^{2} - 9 n + 7 = 0$ .
Answer Type	Text solution:1
Upvotes	150