If-the-third-term-in-the-expansion-of-1-x-x-log-10-x-5-is-1000-then-the-value-of-x-is-

Question Number 55786 by gunawan last updated on 04/Mar/19

If the third term in the expansion of

{(\frac{1}{x} + x^{\log_{10} x})}^{5} is 1000, then the value of

x is

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19

5 c_{2} {(x)}^{3} {(x^{{l o g}_{10} x})}^{2} = 10^{3}

x^{3 + 2 {l o g}_{10} x} = \frac{1000}{5!} \times 2! \times 3!

x^{3 + {l o g}_{10} x^{2}} = \frac{1000 \times 2 \times 3 \times 2}{5 \times 4 \times 3 \times 2} = 10^{2}

{(10^{n})}^{3 + {l o g}_{10} 10^{2 n}} = 10^{2}

{(10^{n})}^{3 + 2 n} = 10^{2}

3 n + 2 n^{2} = 2

2 n^{2} + 3 n - 2 = 0

2 n^{2} + 4 n - n - 2 = 0

2 n (n + 2) - 1 (n + 2) = 0

(n + 2) (2 n - 1) = 0

n = - 2 a n d n = \frac{1}{2}

s o x = 10^{- 2} = \frac{1}{100}

o r x = 10^{\frac{1}{2}} = \sqrt{10}