Question

The centered pentagonal numbers are formed much like the triangular numbers, which we saw in class on Monday. We start in the center with just one dot and then move outward. Define P_n as the sum of the first n pentagons of dots. 1) What are the first five numbers in this sequence? Let's call them: P_1, P_2, P_3, P_4, P_5 (P_4 is pictured to the right) FACT - the formula for the n^th centered pentagonal number is: P_n = (5n^2 - 5n + 2) / 2 2) Check that this formula works for your answers to part (1). 3) Find a recursive formula for P_n. Explain how your answer makes sense in the context of the picture... not just the numbers themselves. 4) Use your answer from part (3) and mathematical induction to prove the formula in part (2).

          The centered pentagonal numbers are formed much like the triangular numbers, which we saw in class on Monday. We start in the center with just one dot and then move outward. Define P_n as the sum of the first n pentagons of dots.

1) What are the first five numbers in this sequence?

Let's call them: P_1, P_2, P_3, P_4, P_5 (P_4 is pictured to the right)

FACT - the formula for the n^th centered pentagonal number is: P_n = (5n^2 - 5n + 2) / 2

2) Check that this formula works for your answers to part (1).

3) Find a recursive formula for P_n. Explain how your answer makes sense in the context of the picture... not just the numbers themselves.

4) Use your answer from part (3) and mathematical induction to prove the formula in part (2).

The centered pentagonal numbers are formed much like the triangular numbers, which we saw in class on Monday. We start in the center with just one dot and then move outward. Define Pn as the sum of the first n pentagons of dots.

1) What are the first five numbers in this sequence?

Let's call them: P1, P2, P3, P4, P5 (P4 is pictured to the right)

FACT - the formula for the n^th centered pentagonal number is: Pn = (5n^2 - 5n + 2) / 2

2) Check that this formula works for your answers to part (1).

3) Find a recursive formula for Pn. Explain how your answer makes sense in the context of the picture... not just the numbers themselves.

4) Use your answer from part (3) and mathematical induction to prove the formula in part (2).

Added by Kimberly P.

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