The product of the first three terms of the increasing equal ratio sequence {an} is 512, and the three terms are subtracted by 1, 3 and 9 respectively to form the equal difference sequence, so as to prove 1 / A1 + 2 / A2 + ﹢n/an﹤1

The product of the first three terms of the increasing equal ratio sequence {an} is 512, and the three terms are subtracted by 1, 3 and 9 respectively to form the equal difference sequence, so as to prove 1 / A1 + 2 / A2 + ﹢n/an﹤1

a1*a2*a3=512 =>(a1*q)^3=512 => a1*q=8 =>a2=8
(a1-1)+(a3-9)=2*(a2-3) => (a1-1)+(a2*q-9)=2*(8-3) => a1+8q=20 =>a2/q+8q=20
The solution is q = 2
a1=4
an=a1*q^(n-1)=4*2^(n-1)=2^(n+1)
Bn=n/An=n/2^(n+1)
Sn=B1+b2+...+Bn
=1/4+2/8+...+(n-1)/2^n+n/2^(n+1)
2Sn=1/2+2/4+...+(n-1)/2^(n-1)+n/2^n
Subtraction has
Sn=(1/2+1/4+1/8+...+1/2^n)-n/2^(n+1)
=(1/2)*[(1/2)^n-1]/(1/2-1)]-n/2^(n+1)
=1-(1/2)^n-n/2^(n+1)