Prove that
sum_(j=0)^n (-(1)/(2))^(j)=(2^(n+1)+(-1)^(n))/(3*2^(n))
whenever n is a nonnegative integer.
13. Prove that 1^(2)-2^(2)+3^(2)-cdots+(-1)^(n-1)n^(2)=(-1)^(n-1) n(n+1)/(2) whenever n is a positive integer.
14. Prove that for every positive integer n,sum_(k=1)^n k2^(k)= (n-1)2^(n+1)+2.
15. Prove that for every positive integer n,
1*2+2*3+cdots+n(n+1)=n(n+1)(n+2)/(3.)
12. Prove that
2(-) 2n+1 + (-1)n 3.2n
whenever n is a nonnegative integer
13. Prove that12 -22 +32 - ...+(-1)n-1n2 =(-1)n-1 n(n + 1)/2 whenever n is a positive integer. 14. Prove that for every positive integer n, E= k2k = (n - 1)2n+1 + 2. 15. Prove that for every positive integer n, 12+23+...+n(n+1=nn+1n+2/3