Binomial Expansion: Find middle term of exp (x^2 + 1/2x)^10 then find x

[M4STR0k]

If the middle term in the expansion of

. . . . .\(\displaystyle \left(x^2\, +\, \dfrac{1}{2x}\right)^{10}\)

...equals 28/27, find the value of x.


Point I'm stuck at:

. . .Middle term = (10 + 2)/2 = 6

. . .\(\displaystyle T_6\, =\, {}^{10} C_5\, x^{10}\, \left(\dfrac{1}{2x}\right)^5\, =\, \dfrac{28}{27}\)

. . .\(\displaystyle T_6\, =\, {}^{10} C_5\, x^{10}\, \dfrac{1}{32x^5}\, =\, \dfrac{28}{27}\)

 

[pka]

[If the middle term in the expansion of

. . . . .\(\displaystyle \left(x^2\, +\, \dfrac{1}{2x}\right)^{10}\)

...equals 28/27, find the value of x.


Point I'm stuck at:

. . .Middle term = (10 + 2)/2 = 6

. . .\(\displaystyle T_6\, =\, {}^{10} C_5\, x^{10}\, \left(\dfrac{1}{2x}\right)^5\, =\, \dfrac{28}{27}\)

. . .\(\displaystyle T_6\, =\, {}^{10} C_5\, x^{10}\, \dfrac{1}{32x^5}\, =\, \dfrac{28}{27}\)

\(\displaystyle \dbinom{10}{5}\dfrac{1}{2^5}=\dfrac{63}{8}\) See the expansion.