Let the coefficients of third, fourth and fifth terms in the expansion of ( x + a x 2 ) n , x 0 , be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to _______.

Question

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 _______.

Smart Achievers · Accepted Answer

(4)

$T_{r + 1} = {}^{n}C_{r} {(x)}^{n - r} {(\frac{a}{x^{2}})}^{r} = {}^{n}C_{r} a^{r} x^{n - 3 r}$

$We have,$ $\frac{{}^{n}C_{2} a^{2}}{{}^{n}C_{3} a^{3}} = \frac{12}{8} \Rightarrow 2 = a (n - 2)$ ...(i)

$and$ $\frac{{}^{n}C_{3} a^{3}}{{}^{n}C_{4} a^{4}} = \frac{8}{3} \Rightarrow 3 = 2 a (n - 3)$ ...(ii)

$Dividing (i) by (ii), we get$

$\frac{2}{3} = \frac{n - 2}{2 (n - 3)} \Rightarrow n = 6$

$From (i),$ $2 = a (6 - 2) \Rightarrow a = \frac{1}{2}$

$For the term independent of x,$ $n - 3 r = 0$

$\Rightarrow$ $6 - 3 r = 0 \Rightarrow r = 2$

$So,$ $T_{3} = {}^{6}C_{2} a^{2} = \frac{15}{4} \approx 4$ $(Approx.)$

Solution

Q.
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 _______.

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Solution

Q. Let the coefficients of third, fourth and fifth terms in the expansion of (x+ax2)n, x≠0, be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to _______.

Ans. (4) Tr+1=Crn(x)n-r(ax2)r=Crn arxn-3r We have, C2n a2C3n a3=128 ⇒ 2=a(n-2) ...(i) and C3n a3C4n a4=83 ⇒ 3=2a(n-3) ...(ii) Dividing (i) by (ii), we get 23=n-22(n-3) ⇒ n=6 From (i), 2=a(6-2) ⇒ a=12 For the term independent of x, n-3r=0 ⇒ 6-3r=0 ⇒ r=2 So, T3=C26 a2=154≈4 (Approx.)

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Q.
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 _______.