Recall the pentagonal numbers: 0 12 Find and prove a closed-form formula for the nth pentagonal number_
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The first pentagonal number is 0, which can be thought of as 0 times the first term (1). Show more…
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Show that a pentagonal number is the sum of a square number and a triangular number. How does the square number relate to the corresponding triangular number? That is, which square number and which triangular number appear in each pentagonal number? Write a recursive formula to represent each pentagonal number in terms of a square number and a triangular number.
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Since we can write a square number as the sum of two consecutive triangular numbers, we can write a pentagonal number in terms of triangular numbers. What is the nth pentagonal number in terms of triangular numbers? Write a recursive formula to represent pentagonal numbers in terms of triangular numbers. Use your results to write a formula to determine the nth pentagonal number in terms of n.
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).
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