Question

If the second, third and fourth term in the expansion of $(x + a)^{n}$ are 240, 720 and 1080 respectively, then the value of n is

• A. 15
• B. 20
• C. 10
• D. 5

Answer 1

Answer: (D)

5

Answer 2

It is given that $T_{2} = 240,T_{3} = 720,T_{4} = 1080$

Now, $T_{2} = 240$ ⇒ $T_{2} =^{n} ⥂ C_{1}x^{n - 1}a^{1} = 240$ .....(i)

and $T_{3} = 720$ ⇒$T_{3} =^{n} ⥂ C_{2}x^{n - 2}a^{2} = 720$ .....(ii)

$T_{4} = 1080$ ⇒ $T_{4} =^{n} ⥂ C_{3}x^{n - 3}a^{3} = 1080$ .....(iii)

To eliminate x, $\frac{T_{2}.T_{4}}{T_{3}^{2}} = \frac{240.1080}{720.720} = \frac{1}{2}$ ⇒ $\frac{T_{2}}{T_{3}}.\frac{T_{4}}{T_{3}} = \frac{1}{2}$.

Now $\frac{T_{r + 1}}{T_{r}} = \frac{n ⥂ C_{r}}{n ⥂ C_{r - 1}} = \frac{n - r + 1}{r}$. Putting $r = 3$ and 2 in above expression, we get $\frac{n - 2}{3}.\frac{2}{n - 1} = \frac{1}{2}$

⇒ $n = 5$