If the coefficients of three consecutive terms in the expansion of (1 + x)n are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

Question

If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

Smart Achievers · Accepted Answer

(4)

$Given coefficients of three consecutive terms are in the ratio 1 : 5 : 20$

$i.e.,$ ${}^{n}C_{r - 1} : {}^{n}C_{r} : {}^{n}C_{r + 1} = 1 : 5 : 20$

$Now,$ $\frac{{}^{n}C_{r - 1}}{{}^{n}C_{r}} = \frac{1}{5} \Rightarrow n = 6 r - 1$ ...(i)

$Also,$ $\frac{{}^{n}C_{r}}{{}^{n}C_{r + 1}} = \frac{5}{20} \Rightarrow n = 5 r + 4$ ...(ii)

$From (i) and (ii), we get$

$6 r - 1 = 5 r + 4 \Rightarrow r = 5 \Rightarrow n = 25 + 4 = 29$

$∴$ $The coefficient of 4^{th} term = {}^{29}C_{3} = 3654$

Solution

Q.
If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

1 1827

2 5481

3 2436

4 3654

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