Let bo, b,, bz, ... be the sequence defined by the explicit formula b, = C- 3" + D(-2)^ for every integer n2 0, where C and D are real numbers. Fill in the blanks in the following proof to show that for any choice of C and D and for each integer k 2 2, b = bk -1+ 6bk – 2° Proof: Let C and D be any real numbers and let k be any integer such that k 2 2. By definition of bo, b,, b,, ..., bx -1 = and bk - 2 = To complete the proof, you need to show that b = b -1+ 6bk - 2. When the expressions for b-1 and b- 2 are substituted into b -1+ 6b - 2, the result is | -«([ bk - 1+ 6b 3k-1 + 6(- Further simplification gives that

Suppose a is a positive integer. Simplify the expression a Σ ((n + 2)³ + n³ - 2(n+1)³). n=1 [Explain in words what you are doing at each step. Your answer should not involve a sum, and your method should not involve evaluating a sum.]

Write the number 3.0292929292929……as an infinite sum; then use the appropriate formula to write it as a quotient of two integers.

find a formula for the sum 2+4+6+..2n when n is a natural number and prove that your formula works