If the fourth term in the expansion of (x+x^_2x)^7 is 4480, then the value of x, where x N…

Question

If the fourth term in the expansion of $\left(x+x^{\log_2x}\right)^7$ is 4480, then the value of $x$, where $x\in N$, is equal to

Accepted Answer

$$T_4=\binom73x^{7-3}\left(x^{\log_2x}\right)^3=35x^4\cdot x^{3\log_2x}=35x^{4+3\log_2x}$$

Given $T_4=4480$: $x^{4+3\log_2x}=128=2^7$

Taking $\log_2$: $(4+3\log_2x)\log_2x=7$

Let $t=\log_2x$: $4t+3t^2=7\Rightarrow3t^2+4t-7=0\Rightarrow(t-1)(3t+7)=0\Rightarrow t=1$ (since $x\in N$)

$$\log_2x=1\quad\Rightarrow\quad x=2$$

Solutions · 1