If the coefficients of the r th term and the (r+1)th term in the expansion of (1+x)20 are in the ratio 1:2, then r is equal to:

Question

If the coefficients of the r th term and the (r+1)th term in the expansion of (1+x)20 are in the ratio 1:2, then r is equal to: (A) 6 (B) 7 (C) 8 (D) 9

Tardigrade Admin · Accepted Answer

Tr=T(r-1)+1=20Cr-1.xr-1 ∴ Coefficient of the r th term =20Cr-1 and coefficient of the (r+1) th term =20Cr ∴ (20Cr-1/20Cr)=(1/2) ⇒ (r/21-r)=(1/2) ⇒ r=7

Q. If the coefficients of the r th term and the $(r + 1) t h$ term in the expansion of $(1 + x)^{20}$ are in the ratio $1 : 2,$ then r is equal to:

6

7

8

9

5

$T_{r} = T_{(r - 1) + 1} =^{20} C_{r - 1} . x^{r - 1}$ $∴$ Coefficient of the r th term $=^{20} C_{r - 1}$ and coefficient of the $(r + 1)$ th term $=^{20} C_{r}$ $∴$ $\frac{^{20} C _{r - 1}}{^{20} C _{r}} = \frac{1}{2}$ $\Rightarrow$ $\frac{r}{21 - r} = \frac{1}{2}$ $\Rightarrow$ $r = 7$

Q. If the coefficients of the r th term and the (r+1)th term in the expansion of (1+x)20 are in the ratio 1:2, then r is equal to:

6

7

8

9

5

Tr​=T(r−1)+1​=20Cr−1​.xr−1 ∴ Coefficient of the r th term =20Cr−1​ and coefficient of the (r+1) th term =20Cr​ ∴ 20Cr​20Cr−1​​=21​ ⇒ 21−rr​=21​ ⇒ r=7

Q. If the coefficients of the r th term and the $(r + 1) t h$ term in the expansion of $(1 + x)^{20}$ are in the ratio $1 : 2,$ then r is equal to:

$T_{r} = T_{(r - 1) + 1} =^{20} C_{r - 1} . x^{r - 1}$ $∴$ Coefficient of the r th term $=^{20} C_{r - 1}$ and coefficient of the $(r + 1)$ th term $=^{20} C_{r}$ $∴$ $\frac{^{20} C _{r - 1}}{^{20} C _{r}} = \frac{1}{2}$ $\Rightarrow$ $\frac{r}{21 - r} = \frac{1}{2}$ $\Rightarrow$ $r = 7$