Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}$, $x \neq 0$, be in the ratio 12:8:3. Then the term independent of x in the expansion is equal to ____.

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Binomial Theorem

Let the coefficients of third, fourth and fifth terms in the expansion of

(x + \frac{a}{x ^{2}})^{n}

x \neq = 0

, be in the ratio 12:8:3. Then the term independent of x in the expansion is equal to ____.

Answer: 4

In the expansion of

(x + \frac{a}{x ^{2}})^{n}

the general term is

T_{r + 1} =^{n} C_{r} a^{r} x^{n - 3 r}

, so the coefficients of the third, fourth and fifth terms are

^{n} C_{2} a^{2}

^{n} C_{3} a^{3}

and

^{n} C_{4} a^{4}

.
From the first pair,

\frac{^{n} C _{2} a ^{2}}{^{n} C _{3} a ^{3}} = \frac{12}{8} ⟹ \frac{3}{( n - 2 ) a} = \frac{3}{2} ⟹ (n - 2) a = 2

From the second pair,

\frac{^{n} C _{3} a ^{3}}{^{n} C _{4} a ^{4}} = \frac{8}{3} ⟹ \frac{4}{( n - 3 ) a} = \frac{8}{3} ⟹ (n - 3) a = \frac{3}{2}

Subtracting the two results gives

a = \frac{1}{2}

, and then

n - 2 = 4

, so

n = 6

.
The term free of

x

needs

n - 3 r = 0

, that is

r = 2

, and it equals

^{6} C_{2} (\frac{1}{2})^{2} = \frac{15}{4} = 3.75

which to the nearest integer is

4

JEE Main · 2021 · 17 Mar 2021 · Shift 2 · Integer answer