Question

Solve the recurrence relation to the basis step. S(1) = 1 S(n) = S(n-1) + (2n-1) for n>= 2 Please explain steps.

Answer 1

To solve the recurrence relation with basis step S(1) = 1, we observe that S(n) = S(n-1) + (2n-1) generates the sequence of square numbers. By proving this pattern through induction or using the known sum of the first n odd numbers equals n^2, we find that S(n) = n^2 is the closed form solution.

Step-by-step explanation:

To solve the recurrence relation S(n) = S(n-1) + (2n-1) with the basis step S(1) = 1, we can use a process to identify a pattern or a closed form for S(n).

• Start with the basis step: S(1) = 1.
• Then apply the recurrence relation repeatedly to find the first few terms:
  • S(2) = S(1) + (2*2-1) = 1 + 3 = 4
  • S(3) = S(2) + (2*3-1) = 4 + 5 = 9
  • S(4) = S(3) + (2*4-1) = 9 + 7 = 16, and so on.
• Observe the sequence: 1, 4, 9, 16, ... which resembles the squares of natural numbers.
• We can guess that S(n) might be n2.
• To prove our hypothesis, we can use induction or look at the sum of the first n odd numbers formula which is known to be equal to n2.

Thus, the closed form solution for the recurrence relation given is S(n) = n2.