等比数列和等差数列各项相乘求和
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等比数列和等差数列各项相乘求和
a1=q
a2=q^2*2
a3=q^3*3
.
aN=q^N×N
求前N项的和
a1=q
a2=q^2*2
a3=q^3*3
.
aN=q^N×N
求前N项的和
sn=a1+a2+a3+……+a(n-1)+an
=q+q^2*2+q^3*3+……+q^(n-1)*(n-1)+q^(n+1)*n
qsn=q^2+q^3*2+q^4*3+……+q^n*(n-1)+q^(n+1)*n
两式相减:
(1-q)sn=q+q^2+q^3+q^4+……+q^n-q^(n+1)*n
=q[1-q^(n+1)]/(1-q)-q^(n+1)*n
=[q-q^(n+1)*n+q^(n+2)*(n-1)]/(1-q)
sn=[q-q^(n+1)*n+q^(n+2)*(n-1)]/(1-q)^2
=q+q^2*2+q^3*3+……+q^(n-1)*(n-1)+q^(n+1)*n
qsn=q^2+q^3*2+q^4*3+……+q^n*(n-1)+q^(n+1)*n
两式相减:
(1-q)sn=q+q^2+q^3+q^4+……+q^n-q^(n+1)*n
=q[1-q^(n+1)]/(1-q)-q^(n+1)*n
=[q-q^(n+1)*n+q^(n+2)*(n-1)]/(1-q)
sn=[q-q^(n+1)*n+q^(n+2)*(n-1)]/(1-q)^2