How can we calculate? (1-x) + (1-x)^2 + (1-x)^3 + (1-x)^4 + (1-x)^5.... + (1-x)^n

1 Answer
Mar 24, 2018

((1 + (1 - x)^n) (1-x))/( x-2)

Explanation:

The polynomial identity

sum_(k=0)^n a^k = (a^(n+1)-1)/(a-1) leave us to

sum_(k=1)^n (1-x)^k =((1-x)^(n+1)-1)/((x-1)-1)-1 = ((1 + (1 - x)^n) (1-x))/( x-2)

For x ne 2

If instead we have

sum_(k=0)^(n-1) (1-x)^k =((1-x)^n-1)/((x-1)-1) = ((1-x)^n-1)/(x-2)

for x ne 2