等比数列{an}的首项a1=-1,前n项和为Sn,若S10/S5=31/32,则S15/S10=?

Question

等比数列{an}的首项a1=-1,前n项和为Sn,若S10/S5=31/32,则S15/S10=?

Answer 1

等比数列{a‹n›}的首项a₁=-1,前n项和为S‹n›,若S₁₀/S₅=31/32,则S₁₅/S₁₀=?
S‹n›=a₁(1-qⁿ)/(1-q),故S₁₀/S₅=[a₁(1-q¹⁰)/(1-q)]/[a₁(1-q⁵)/(1-q)]=(1-q¹⁰)/(1-q⁵)
=(1+q⁵)(1-q⁵)/(1-q⁵)=1+q⁵＝31/32,q⁵＝31/32-1=-1/32,∴q=-1/2；
故S₁₅/S₁₀=（1-q¹⁵)/(1-q¹⁰)=(1-q⁵)(1+q⁵+q¹⁰)/(1+q⁵)(1-q⁵)=)(1+q⁵+q¹⁰)/(1+q⁵)
=1+(q¹⁰)/(1+q⁵)=1+(q⁵)/[(1/q⁵)+1]=1+(-1/2)⁵/[(-2)⁵+1]=1+(-1/32)/[-32+1)]=1+(1/992)=993/992