Question: N-1 - () = 1-2 n-1 1. (27 points) a) Compute the following sums (10 points): S, =1+3+5+7+ ... +79 S, =1+2+4+8+16+ ... +1024 S,

N-1 - () = 1-2 n-1 1. (27 points) a) Compute

N-1 - () = 1-2 n-1 1. (27 points) a) Compute the following sums (10 points): S, =1+3+5+7+ ... +79 S, =1+2+4+8+16+ ... +1024 S, =7i S, =(2i -1) S, =4*1 b) Find the order of growth of the following sums. Use the @[g(n)) notation with the simplest function g(n) possible (12 points): 5(ij) i=0 j=i c) Prove the formula (5 points): i = n(n+1)/2 i=1 either by mathematical induction or by following the insight of a 10- year-old school boy named Carl Friedrich Gauss (17771855) who grew up to become one of the greatest mathematicians of all times. (iP+n) Iogi" " i2"+1 i=0 i=2 i=1

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