GP members

The geometric sequence has 10 members. The last two members are 2 and -1. Which member is -1/16?

Result

x =  14

Solution:

a9=2 a10=−1  q=a10/a9=(−1)/2=−12=−0.5  a(n)=a1 q(n−1)  a8=a9/q=2/(−0.5)=−4  a11=q⋅ a10=(−0.5)⋅ (−1)=12=0.5 a12=q⋅ a11=(−0.5)⋅ 0.5=−14=−0.25 a13=q⋅ a12=(−0.5)⋅ (−0.25)=18=0.125 a14=q⋅ a13=(−0.5)⋅ 0.125=−116=−0.0625  x=14 x>10??a_{ 9 } = 2 \ \\ a_{ 10 } = -1 \ \\ \ \\ q = a_{ 10 }/a_{ 9 } = (-1)/2 = - \dfrac{ 1 }{ 2 } = -0.5 \ \\ \ \\ a(n) = a_{ 1 } \ q^(n-1) \ \\ \ \\ a_{ 8 } = a_{ 9 }/q = 2/(-0.5) = -4 \ \\ \ \\ a_{ 11 } = q \cdot \ a_{ 10 } = (-0.5) \cdot \ (-1) = \dfrac{ 1 }{ 2 } = 0.5 \ \\ a_{ 12 } = q \cdot \ a_{ 11 } = (-0.5) \cdot \ 0.5 = - \dfrac{ 1 }{ 4 } = -0.25 \ \\ a_{ 13 } = q \cdot \ a_{ 12 } = (-0.5) \cdot \ (-0.25) = \dfrac{ 1 }{ 8 } = 0.125 \ \\ a_{ 14 } = q \cdot \ a_{ 13 } = (-0.5) \cdot \ 0.125 = - \dfrac{ 1 }{ 16 } = -0.0625 \ \\ \ \\ x = 14 \ \\ x>10 ??

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