Question

if the coeffcient of r th term and (r+4)th term are equal in the expansion of (1+x)^20 , then the value of r will be
(a)7
(b)8
(c)9
(d)10

Answer 1

Answer:

Option ( c ) = 9

Step-by-step explanation:

Given equation = ( 1 + x )²⁰

⇒ a = 1, b = x, n = 20

$T_{r+1} = \: ^nC_r.\:\:a^{n-r}.\:\:b^r\\\\\text{Substituting all the values we get,}\\\\T_{r+1} = \:^{20}C_r.\:\:1^{n-r}.\:\: x^r\\\\T_{r+1} = \:^{20}C_r.\:\: x^{r}$

Case 1 : To find the coefficient of r th term:

Substituting r = r - 1, we get,

$T_{r - 1 + 1 } = \: ^{20}C_{r -1 }.\:\: x^{r-1}\\\\T_{r-1} = \:^{20}C_r.\:\:x^{r-1}$

So coefficient is $^{20}C_{r-1}$

Case 2: To find the coefficient of r + 4 th term

Substituting r = r + 3 we get,

$T_{r+3+1} = \:^{20}C_{r+3}.\:\:x^{r+3}\\\\T_{r+4} = \:^{20}C_{r+3}.\:\:x^{r+3}$

Hence coefficient is $^{20}C_{r+3}$

Now it is given that the coefficients are equal. Hence we get,

$^{20}C_{r-1} = ^{20}C_{r+3}$

We know that if,

$^{n}C_{r} = ^{n}C_{p}, \:\: \text{Then r = n - p }$

Since in our solution we get both n to be equal we can write the equation as:

⇒ r - 1 = 20 - ( r + 3 )

⇒ r - 1 = 20 - r - 3

⇒ r - 1 = 17 - r

⇒ r + r = 17 + 1

⇒ 2r = 18

⇒ r = 18/2

⇒ r = 9

Hence 9 is the answer.