Question

Use Expand, Guess, and Verify to find a closed form solution for

the following recurrence relation:
T(n) = 3T(n-1)
T(1) = 3

Answer 1

The Expand, Guess, and Verify method leads to a closed form solution for the recurrence relation T(n) = 3T(n-1) with T(1) = 3, which is T(n) = 3^n.

Step-by-step explanation:

The question is asking for a closed form solution for the recurrence relation T(n) = 3T(n-1), with the base case T(1) = 3 using the Expand, Guess, and Verify method. We begin by expanding the recurrence:

• T(n) = 3T(n-1)
• T(n-1) = 3T(n-2)
• T(n-2) = 3T(n-3)
• ...
• T(2) = 3T(1)

Now, we substitute back into our original equation iteratively:

• T(n) = 3(3T(n-2))
• T(n) = 3^2T(n-2)
• T(n) = 3^2(3T(n-3))
• T(n) = 3^3T(n-3)
• ...
• T(n) = 3^(n-1)T(1)

Using our base case, T(1) = 3, we can then guess that T(n) = 3^n. To verify this, we can see that it holds for the base case, and assuming T(k) = 3^k, then T(k+1) = 3T(k) = 3(3^k) = 3^(k+1), which maintains the pattern. Therefore, the closed form solution is T(n) = 3^n.