Question

The two successive terms in the expansion of (1 + x)²⁴ whose coefficients are in the ratio 1:4 are ______.

Options

• 3^rd and 4^th

• 4^th and 5^th

• 5^th and 6^th

• 6^th and 7^th

Answer 1

The two successive terms in the expansion of (1 + x)²⁴ whose coefficients are in the ratio 1:4 are 5^th and 6^th.

Explanation:

Let r^th and (r + 1)^th be two successive terms in the expansion (1 + x)²⁴

∴ ${\displaystyle {\text{T}}_{r+1}={}^{24}{\text{C}}_r\cdot x^r}$

${\displaystyle {\text{T}}_{r+2}={\text{T}}_{r+1+1}={}^{24}{\text{C}}_{r+1}{x}^{r+1}}$

 We have ${\displaystyle \frac{{}^{24}{\text{C}}_r}{{}^{24}{\text{C}}_{r+1}}=\frac{1}{4}}$

⇒ ${\displaystyle \frac{\frac{24!}{r!(24-r)!}}{\frac{24!}{(r+1)!(24-r-1)!}}=\frac{1}{4}}$

⇒ ${\displaystyle \frac{24!}{r!(24-r)!}\times \frac{(r-1)!(24-r-1)!}{24!}=\frac{1}{4}}$

⇒ ${\displaystyle \frac{(r+1)\cdot r!(24-r-1)!}{r!(24-r)(24-r-1)!}=\frac{1}{4}}$

⇒ ${\displaystyle \frac{r+1}{24-r}=\frac{1}{4}}$

⇒ 4r + 4 = 24 – r

⇒ 5r = 20

⇒ r = 4

∴ T₄₊₁ = T₅ and T₄₊₂ = T₆