Find the value of x for which thr middle term in the expansion of ( 1 + x ) ^ 20 in ascending powers of x is the geometric mean of the 9th and the 12th terms.

Answer 1

the middle term of the expansion is term(10)
term(10) = C(20,9)(1)^11 x^9
= 167960x^9

the 9th term is
C(20,8) x^8 = 125970x^8
the 12 term is C(20,11) x^11 = 167960x^11

geometric mean = √(125970(167960)x^19)
167960x^9 = √(125970(167960)x^19)
167960x^9 = √(125970(167960))*x^9 √x
167960 = √(125970(167960) √x
square both sides
167960^2 = 125970(167960)x

I got this to reduce to x = 4/3

check:
middle term = 2236940.819 , using my calculator
term9 = 1258279.211
term12 = 3976783.676
G-mean = 2236940.819

yeahhhh!!!

Answer 2

Hmmm. Since there are 21 terms in the expansion, the 11th term is the middle one.

Adjust the numbers accordingly.