正项等比数列中a7a9+a7a6+a8a9=16则S14

Question

正项等比数列中 a7a9+a7a6+a8a9=16则S14

Answer 1

a(n) = ar^(n-1),n = 1,2,...
16 = a(7)a(9) + a(7)a(6) + a(8)a(9) = a^2r^14 + a^2r^11 + a^2r^15
= a^2r^10[r^4 + r + r^5],
r^4 + r + r^5 > 0.
a = 4/|r^5(r^4 + r + r^5)^(1/2)| 或 a = -4/|r^5(r^4 + r + r^5)^(1/2)|
若r = 1,则,16 = 3a^2,a^2 = 16/3,a = 4/3^(1/2)或a = -4/3^(1/2).
S(n) = na,
S(14) = 14*4/3^(1/2)或S(14) = -14*4/3^(1/2).
若r 不等于1,则S(n) = a[r^n - 1]/(r-1).
S(14) = a[r^14 - 1]/(r - 1) = 4[r^14 - 1]/[(r-1)|r^5(r^4 + r + r^5)^(1/2)|]
或
S(14) = a[r^14 - 1]/(r - 1) = -4[r^14 - 1]/[(r-1)|r^5(r^4 + r + r^5)^(1/2)|]