Use Python for the programs asked in b, c, and d.

We consider the recurrence x_n + 1 = 13/3 x_n - 4/3 x_n - 1 (for all n greaterthanorequalto 1), when x_0,x_1 are given. Prove by induction that if x_0 = 1 and x_1 = 1/3, then x_n = 3^-n for all n greaterthanorequalto 0. Compute 3^-n for n = 1 through 50, in double precision, by directly using the formula 3^-n (in a for-loop, in Python or in C). Give the resulting 50 numbers. Compute x_n by using the recurrence equation in (a) in a small program, for n = 1 through 50. Use double precision. Give the resulting 50 numbers, and your program. Use the recurrence equation in (a), but now with x_0 = 1 and x_1 = 1/3 - 10^-4, to compute x_n for n = 1 through 50. Give the resulting 50 numbers, and your program. Find two numbers A and B such that for all n greaterthanorequalto 0: x_n = 3^-n A + 4^n B, with x_0 = 1, x_1 = 1/3 - 10^-4. In light of (e), explain the events in (c). (Think in terms of amplification and propagation of numerical imprecision, or "errors".)