Could someone please check my work

Prove that Itrest... to" = lopat! 1 -r for all nell, when rel. Let P (n) be the statement i=0 1 -5 - for all nonnegative integers, n, when r+ 1. Then PCI) asserts that LHS = EriEr+ralts =1-rit1 1 -r = 1-r2 = (1-r) (its) 1 -r = 1+r = RHS LHS =1 = PHS Thus, PCI) is True, which establishes The basis for inducrite Assume That Epi = lopkel is True for some nonnegeri integer, K, when ral. 1+1 WTS r = 1 - K+ + 1 lark+2 1 -r K+1 LHS = = Use the assumption = 1-r -141 + - R+1 1 -r = 1 - - k+1 +r ( 1 - 8) = 1- k+1 K+ 1 + r - r - R+ 2 1 -r = 1 -r k+ 2 = RHS Thus, P (n ) = 1 -5 + 1 for all nonnegative integers, n, when ral