S = r^0 + r^1 + r^2 + r^3 + ... + r^(n - 1)

S * r = r^1 + r^2 + r^3 + ... + r^(n - 1) + r^n

Sr - S = r^(n) - r^0

S * (r - 1) = r^n - 1

S = (1 - r^n) / (1 - r)

let u= 1+r...r^(n-1)

ru=r(1+r...r^(n-1)=r+r^2...r^n

ru-u=(0-1)+(r-r)+(r^2-r^2)...r^n every term cept the first and last were able to pair up

ru-u=-1+r^n

u(r-1)=r^n-1

u=(r^n-1)/(r-1) then they just multiply both top and bottom by -1 not changing value

u=(1-r^n)/(1-r)

? = a million + r + r² + ... +r^(n-a million) r? =? ? r + r² + ... + r^(n-a million) + r^n subtract eq2 from eq1 ? - r? = a million - r^n (a million - r) ? = a million - r^n ? = (a million - r^n) / (a million - r) granted, it is deduction, no longer induction.