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

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Answer 1 · 2018-03-24T14:42:10

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

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#