Question

Let {an​}n=1[infinity]​ be a recursively defined sequence with initial conditions a1​=2 and recurrence relation an​=7+an−1​. Guess a closed formula for this sequence (be sure to show or somehow indicate how you arrived at your answer) Let {vn​}n=1[infinity]​ be a recursively defined sequence with initial conditions v1​=1 and recurrence relation vn​=vn−1​+2n. Guess a closed formula for this sequence (be sure to show or somehow indicate how you arrived at your answer). If your answer uses sigma notation, this is fine. Let {an​}n=0[infinity]​ be a recursively defined sequence with initial conditions a0​=4 and recurrence relation an​=9+an−1​. Prove that an​=4+9n for all n≥0. Let {vn​}n=1[infinity]​ be a recursively defined sequence with initial conditions v1​=1 and recurrence relation vn​=vn−1​+2n. Guess a closed formula for this

Answer 1

The closed-form expressions for the recursively defined sequences are an = 7n - 5 for the first, and vn = n2 + n for the second, with the proof for an = 4 + 9n using induction for the third sequence.

Step-by-step explanation:

The task is to find closed-form expressions for given recursively defined sequences and to prove an assertion about another recursively defined sequence.

For the sequence {an} defined by the initial condition a1 = 2 and the recurrence relation an = 7 + an-1, we can determine a closed-form expression by observing the first few terms: a2 = 7 + a1 = 9, a3 = 7 + a2 = 16, and so on. Noticing the pattern, the closed-form expression is an = 7n - 5.

For the sequence {vn} defined by the initial condition v1 = 1 and the recurrence relation vn = vn-1 + 2n, we analyze the first few terms and search for a pattern. Using the formula for the sum of the first n natural numbers and manipulating it to fit the pattern of the given sequence, we deduce a closed-form expression vn = n2 + n.

Lastly, we prove for the sequence {an} with the initial condition a0 = 4 and the recurrence relation an = 9 + an-1 that the expression is indeed an = 4 + 9n for all n ≥ 0 using induction.