8. 1+5+9+13+……+(4n-3) = n(2n-1) P 1 = 1(2(1)-1)=1 check.

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……+(4n-3) = n(2n-1) P 1 = 1(2(1)-1)=1 check

……+(4n-3) = n(2n-1) P 1 = 1(2(1)-1)=1 check S k = k(2k-1) S k+1 = (k+1)(2(k+1)-1) = (k+1)(2k+1) a k+1 = 4(k+1)-3 = 4k+1

……+(4n-3) = n(2n-1) P 1 = 1(2(1)-1)=1 check S k = k(2k-1) S k+1 = (k+1)(2(k+1)-1) = (k+1)(2k+1) a k+1 = 4(k+1)-3 = 4k+1 Proof S K+1 = S k + a k+1 (k+1)(2k+1) = k(2k-1) + 4k+1

……+(4n-3) = n(2n-1) P 1 = 1(2(1)-1)=1 check S k = k(2k-1) S k+1 = (k+1)(2(k+1)-1) = (k+1)(2k+1) a k+1 = 4(k+1)-3 = 4k+1 Proof S K+1 = S k + a k+1 (k+1)(2k+1) = k(2k-1) + 4k+1 =2k 2 -k+4k+1= 2k 2 +3k+1 = (2k+1)(k+1) check

…..+n 3 = n 2 (n+1) 2 4 P 1 = 1 2 (1+1) 2 = 1 check 4

…..+n 3 = n 2 (n+1) 2 4 P 1 = 1 2 (1+1) 2 = 1 check 4 S k = k 2 (k+1) 2 /4 S k+1 = (k+1) 2 ((k+1)+1) 2 /4 = ((k 2 +2k+1)(k 2 +4k+4))/4 a k+1 = (k+1) 3 Proof S k+1 = S k + a k+1

14. (Cont.) Proof S k+1 = S k + a k+1 [(k 2 +2k+1)(k 2 +4k+4)]/4 = k 2 (k+1) 2 /4 + (k+1) 3

14. (cont.) Proof S k+1 = S k + a k+1 [(k 2 +2k+1)(k 2 +4k+4)]/4 = k 2 (k+1) 2 /4 + (k+1) 3 = [k 2 (k+1) 2 + 4(k+1) 3 ]/4 = k 2 (k 2 +2k+1) + 4(k+1)(k 2 +2k+1)/4

Proof S k+1 = S k + a k+1 [(k 2 +2k+1)(k 2 +4k+4)]/4 = k 2 (k+1) 2 /4 + (k+1) 3 = [k 2 (k+1) 2 + 4(k+1) 3 ]/4 = k 2 (k 2 +2k+1) + 4(k+1)(k 2 +2k+1)/4 = (k 2 +2k+1)(k 2 +4k+4)/4 check