JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Binomial Theorem

If the coefficients of the 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

Answer: (A)

Solution

Let the three consecutive terms be

T_{r}, T_{r + 1}, T_{r + 2}

with coefficients

^{n} C_{r - 1},^{n} C_{r},^{n} C_{r + 1}

.

Given ratio:

\frac{^{n} C _{r - 1}}{^{n} C _{r}} = \frac{1}{5}

and

\frac{^{n} C _{r}}{^{n} C _{r + 1}} = \frac{1}{4}

.

Using the property

\frac{^{n} C _{r - 1}}{^{n} C _{r}} = \frac{r}{n - r + 1}

, we get:

\frac{r}{n - r + 1} = \frac{1}{5}

⟹

5 r = n - r + 1

⟹

6 r = n + 1

Similarly,

\frac{r + 1}{n - r} = \frac{1}{4}

⟹

4 (r + 1) = n - r

⟹

5 r = n - 4

Solving:

6 r = n + 1

and

5 r = n - 4

gives

r = 5

and

n = 29

.

The fourth term has coefficient

^{29} C_{3} = \frac{29 \times 28 \times 27}{6} = 3654

JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Single correct option