Math Problem Statement

Consider the following recurrence relation. P(n) =

1 if n = 0 P(n − 1) + n2 if n > 0 (a) Compute the first eight values of P(n). (Enter your answers as a comma-separated list of values ordered with respect to increasing n.)

Solution

Given the recurrence relation:

1 & \text{if } n = 0 \\ P(n - 1) + n^2 & \text{if } n > 0 \end{cases}$$ Let's compute the first eight values of $$P(n)$$ for $$n = 0, 1, 2, \dots, 7$$. ### Step-by-step calculations: - $$P(0) = 1$$ - $$P(1) = P(0) + 1^2 = 1 + 1 = 2$$ - $$P(2) = P(1) + 2^2 = 2 + 4 = 6$$ - $$P(3) = P(2) + 3^2 = 6 + 9 = 15$$ - $$P(4) = P(3) + 4^2 = 15 + 16 = 31$$ - $$P(5) = P(4) + 5^2 = 31 + 25 = 56$$ - $$P(6) = P(5) + 6^2 = 56 + 36 = 92$$ - $$P(7) = P(6) + 7^2 = 92 + 49 = 141$$ Thus, the first eight values of $$P(n)$$ are: $$1, 2, 6, 15, 31, 56, 92, 141$$ Let me know if you'd like more details or have any questions! Here are 5 related questions: 1. How can recurrence relations be solved using closed-form expressions? 2. What are some real-world applications of recurrence relations? 3. Can you derive a general formula for $$P(n)$$ from this recurrence relation? 4. How does this recurrence relate to other well-known sequences, such as the triangular numbers? 5. What are the different methods to approach solving recurrence relations? **Tip**: When solving recurrence relations, try expanding a few terms to identify patterns that can help with simplifications.

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Math Problem Analysis

Mathematical Concepts

Recurrence Relations
Algebra
Summation

Formulas

P(n) = P(n-1) + n^2
Base Case: P(0) = 1

Theorems

Recurrence relation definition

Suitable Grade Level

Grades 9-12