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

∴ `"T"_(r + 1) = ""^24"C"_r * x^r`

`"T"_(r + 2) = "T"_(r + 1 + 1) = ""^24"C"_(r + 1) x^(r + 1)`

 We have `(""^24"C"_r)/(""^24"C"_(r + 1)) = 1/4`

⇒ `((24!)/(r!(24 - r)!))/((24!)/((r + 1)!(24 - r - 1)!)) = 1/4`

⇒ `(24!)/(r!(24 - r)!) xx ((r - 1)!(24 - r - 1)!)/(24!) = 1/4`

⇒ `((r + 1) * r!(24 - r - 1)!)/(r!(24 - r)(24 - r - 1)!) = 1/4`

⇒ `(r + 1)/(24 - r) = 1/4`

⇒ 4r + 4 = 24 – r

⇒ 5r = 20

⇒ r = 4

∴ T₄₊₁ = T₅ and T₄₊₂ = T₆