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

Question

If the coeff of $2^{n d}, 3^{r d}$ and the $4^{t h}$ terms in the expansion of $(1 + x)^{n}$ are in A.P, then the value of $n$ is :

2

7

11

14

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James Discussed

If the coeff of

2^{n d}, 3^{r d}

and the

4^{t h}

terms in the expansion of

(1 + x)^{n}

are in A.P, then the value of

n

is :

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Text solutionVerified

Solution:

We know that in the coefficient of

r^{t h}

term in expansion of

(1 + x)^{n} =^{n} C_{r - 1}

Now, Coefficient of

2^{n d}

term in the expansion

(1 + x)^{n} =^{n} C_{1}

Coefficient of

3^{r d}

term in the expansion

(1 + x)^{n} =^{n} C_{2}

Coefficient of

4^{t h}

term in the expansion

(1 + x)^{n} =^{n} C_{3}

Since,

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

and

^{n} C_{3}

are in A.P

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

or,

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

or,

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

or,

6 + (n - 1) (n - 2) = 6 (n - 1)

or,

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

or,

n^{2} - 9 n + 14 = 0

or,

(n - 7) (n - 2) = 0

or,

n = 7

n = 2

If we take

n = 2

then number of terms in the expansion are

3

which can't satisfy the above given condition. Therefore

n = 7

is the solution.

Hence, B is the correct option.

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Question Text	If the coeff of $2^{n d}, 3^{r d}$ and the $4^{t h}$ terms in the expansion of $(1 + x)^{n}$ are in A.P, then the value of $n$ is :
Answer Type	Text solution:1
Upvotes	150

If the coeff of 2nd,3rd and the 4th terms in the expansion of (1+x)n are in A.P, then the value of n is :

Text solutionVerified

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If the coeff of $2^{n d}, 3^{r d}$ and the $4^{t h}$ terms in the expansion of $(1 + x)^{n}$ are in A.P, then the value of $n$ is :