Question

Find the 7th term in the expansion of [ 4x/5 + 5/2x ]8

Answer 1

Answer:4375/x^4

T(6+1)=8C6(4x/5)^8-6(6/2x)^6

8×7×6!/6!×2×1(4x/5)^2(5/2x)^6

28×16x^2/25×15625/64x^6

4375/x4

Answer 2

$\underline {By \: Binomial \: Expansion: }}$

$In \: (x+a)^{n} \: expansion$

$n^{th} \:term = t_{r+1} = ^{n}C_{r} \times x^{n-r}\times a^{r}$

$Given \: \left( \frac{4x}{5} + \frac{5}{2x}\right)^{8}$

$Here , \: x = \frac{4x}{5} , \: a = \frac{5}{2x},\\n = 8 , \: and \: r = 6$

$7^{th} \:term \\= t_{6+1} \\= ^{8}C_{6} \times \left( \frac{4x}{5} \right)^{8-6} \times \left( \frac{5}{2x} \right)^{6}\\= \frac{8!}{(8-6)! 6!} \times \left( \frac{4x}{5} \right)^{2} \times \left( \frac{5}{2x} \right)^{6}$

$= 28 \times \frac{16x^{2}}{5^{2}} \times \frac{5^{6}}{64x^{6}}\\= \frac{7 \times 5^{4}}{x^{4}}$

$= \frac{4375}{x^{4}}$

Therefore.,

$\red { 7^{th} \:term} \green {= \frac{4375}{x^{4}} }$

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