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In the expansion (1+x)^n if the coefficient of the 8th term is

Question

In the expansion $(1 + x)^{n}$ if the coefficient of the 8th term is 4 times the coefficient of the 9th term, then find the total number of terms in the expansion.

none of these

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In the expansion

(1 + x)^{n}

if the coefficient of the 8th term is 4 times the coefficient of the 9th term, then find the total number of terms in the expansion.

15 mins ago

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Text solutionVerified

Given,

(1 + x)^{n}

General term,

T_{r + 1} =^{n} C_{r} x^{n - r} y^{r}

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

T_{8} = T_{7 + 1} =^{n} C_{7} (1)^{n - 7} (x)^{7}

T_{9} = T_{8 + 1} =^{n} C_{8} (1)^{n - 8} (x)^{8}

Given, the coefficient of

8

th term is

4

times coefficient of

9

th term

^{n} C_{7} = 4^{n} C_{8}

\frac{n !}{7 ! ( n - 7 )!} = 4 \times \frac{n !}{8 ! ( n - 8 )!}

.....................

n! = n (n - 1)!

8! (n - 8)! = 4 \times 7! (n - 7)!

8 \times 7! (n - 8)! = 4 \times 7! \times (n - 7) (n - 8)!

8 = 4 (n - 7)

2 = n - 7

n = 9

Hence, the total number of terms in the expansion are

9

Hence, correct option is A.

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Question Text	In the expansion $(1 + x)^{n}$ if the coefficient of the 8th term is 4 times the coefficient of the 9th term, then find the total number of terms in the expansion.
Answer Type	Text solution:1
Upvotes	150