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Question
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
Options
3rd and 4th
4th and 5th
5th and 6th
6th and 7th
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Solution
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are 5th and 6th.
Explanation:
Let rth 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
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