D1. Prove by induction that for all natural numbers n - - n (c ) (a) 1+3+5+ ... +(2n - 1) = n2; (b) (1)(2) + (2)(3) +
Posted: Thu May 12, 2022 1:27 pm
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- - n (c ) (a) 1+3+5+ ... +(2n - 1) = n2; (b) (1)(2) + (2)(3) + ... +(n)(n+1) = n(n + 1)(n+2). b 3 1 1 1 + + t...t + (1)(2) (2)(3) (n)(n + 1) n+1' 1 1 1 (d) + +. + (1)(3)(3)(5) (2n - 1)(2n + 1) 2n +1 na +n is divisible by 2 (Can you prove this directly?); n! + 2n is divisible by 3; (g) a" – bn is divisible by a – b (Hint: Show that ak+1 – 66+1 = a (a - b) + b(ač – bk)). g n . . . (e) (f) - = - -