Prove that 3 + (3 times 5) + (3 times 5^2) + .. + (3 times 5^n) = (3 times (5^n + 1 - 1)/4 whenever n is a nonnegative integer. a) Find a formula for 1/2 + 1/4 + 1/8 + .. + 1/2^n by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part (a). Prove that for every positive integer n, (1 times 2) + (2 times 3) +. + (n times(n + 1)) = (n times(n + 1) times (n + 2))/3 Which amounts of money can be formed using just two-dollar bills and five-dollar bills? Prove your answer using strong induction. Find f(2), f (3), and f (4) if f is defined recursively by f(0) = -1, f(1) = 2, and for n = 1, 2, .. a) f(n + 1) = f(n) + 3f(n - 1) b) f(n + 1) = (f(n)^2) times (f(n - 1)) Assume f_n is the nth Fibonacci number. Prove that f_1 + f_3 +. + f_2n - 1 = f_2n when n is a positive integer