Coefficient Party

Probability Level 3

$\left ( x^{- \frac 8 3} +x^2 \ \log_{10}x \right )^8$

For what value of $x$ is the sixth term in the binomial expansion above equal to 5600?

Details and Assumptions

The first few term of the binomial expansion $(a+b)^n$ is ${n \choose 0} a^n, {n \choose 1} a^{n-1} b, {n \choose 2} a^{n-2} b^2, \ldots$

1 solution

Pranjal Jain
Dec 27, 2014

Using binomial expansion,

$6^{th}$ term $=\binom{8}{3}(x^\frac{-8}{3})^3(x^2×\log_{10} x)^5=5600\\\Rightarrow 56x^{-8}×(x^2×log_{10} x)^5=5600\\\Rightarrow x^2\log_{10} x=100$

By observation, $x=\boxed{10}$