The-pisitive-value-of-a-so-that-the-coefficient-of-x-5-and-x-15-are-equal-in-the-expansion-of-x-2-a-x-3-10-

Question Number 56138 by gunawan last updated on 11/Mar/19

The pisitive value of a so that the

coefficient of x^{5} and x^{15} are equal in the

expansion of {(x^{2} + \frac{a}{x^{3}})}^{10}

Commented by maxmathsup by imad last updated on 11/Mar/19

w e h a v e {(x^{2} + \frac{a}{x^{3}})}^{10} = \sum_{k = 0}^{10} C_{10}^{k} {(x^{2})}^{k} {({a x}^{- 3})}^{10 - k}

= \sum_{k = 0}^{10} C_{10}^{k} x^{2 k} a^{10 - k} x^{- 30 + 3 k} = \sum_{k = 0}^{10} C_{10}^{k} a^{10 - k} x^{5 k - 30}

t h e c o e f f i c i e n t o f x^{5} i s λ_{5} / 5 k - 30 = 5 \Rightarrow k = 7 \Rightarrow

λ_{5} = C_{10}^{7} a^{3} a l s o t h e c o e f g i c i e n t o f x^{15} i s λ_{15} / 5 k - 30 = 15 \Rightarrow 5 k = 45 \Rightarrow k = 9

\Rightarrow λ_{15} = C_{10}^{9} a s o λ_{5} = λ_{15} \Rightarrow C_{10}^{7} a^{3} = C_{10}^{9} a \Rightarrow

\frac{10!}{7! 3!} a^{3} = 10 a \Rightarrow \frac{10 . 9.8}{6} a^{3} = 10 a \Rightarrow \frac{3.3.4.2}{3.2} a^{3} = a \Rightarrow 12 a^{3} - a = 0 \Rightarrow

a (12 a^{2} - 1) = 0 \Rightarrow a = 0 o r 12 a^{2} = 1 \Rightarrow a = 0 o r a = \bar{+} \frac{1}{2 \sqrt{3}} .

Commented by maxmathsup by imad last updated on 11/Mar/19

b u t a > 0 \Rightarrow a = \frac{1}{2 \sqrt{3}} .

Answered by tanmay.chaudhury50@gmail.com last updated on 11/Mar/19

r + 1 t h t e r m c o n t a i n s x^{5}

10 c_{r} {(x^{2})}^{10 - r} {(\frac{a}{x^{3}})}^{r}

= 10 c_{r} \times x^{20 - 2 r - 3 r} \times a^{r}

g i v e n 20 - 5 r = 5

5 r = 15 \to r = 3

10 c_{3} \times x^{20 - 5 \times 3} \times a^{3} \to \frac{10!}{3! 7!} \times a^{3} \times x^{5}

\dots

r_{1} + 1 t h t e r m c o n t s i n s x^{15}

10 c_{r_{1}} \times x^{20 - 5 r_{1}} \times a^{r_{1}}

20 - 5 r_{1} = 15 \to 5 r_{1} = 5 r_{1} = 1

10 c_{1} \times x^{15} \times a^{1} \to \frac{10!}{1! 9!} \times a^{1} \times x^{15}

a c c o r d i n g t o q u e s t i o n

\frac{10!}{3! 7!} a^{3} = \frac{10!}{1! 9!} a^{1}

a^{2} = \frac{3 \times 2 \times 7!}{9 \times 8 \times 7!} = \frac{1}{3 \times 4}

a = \frac{1}{2 \sqrt{3}}