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magic88
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Homework Statement
This is a proof problem in the mathematical induction section of my textbook. I am having trouble with question (c).
Result: 12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/6 for every positive integer n.
(a) Use the result to determine the formula for 22 + 42 + 62 + ... + (2n)2 for every positive integer n.
- I have already found the solution to this problem: 22 + 42 + 62 + ... + (2n)2 = (2/3)n(n+1)(2n+1) for every positive integer n.
(b) Use the result to determine the formula for 12 + 32 + 52 + ... + (2n -1)2 for every positive integer n.
- I have already found the solution to this problem: 12 + 32 + 52 + ... + (2n -1)2 = (2/3)n(n+1)(2n+1) - 2n(n+1) + n for every positive integer n.
(c) Use (a) and (b) to determine the formula for 12 - 22 + 32 - 42 + ... + (-1)n+1)n2 for every positive integer n. Use mathematical induction to prove this.
Homework Equations
Principle of Mathematical Induction: For each positive integer n, let P(n)be a statement. If P(1) is true and the implication "If P(k), then P(k+1) is true," then P(n) is true for every positive integer n.
The Attempt at a Solution
I have found the solutions to part (a) and (b) and have proved them. (Lots of algebra involved!) But I've been trying to figure out part (c) for about an hour now, and I just don't know how to approach it.
I started by listing out a few of the sums from part (c):
- When n = 1, sum = 1
- When n = 2, sum = -3
- When n = 3, sum = 6
- When n = 4, sum = -10
- When n = 5, sum = 15
- When n = 6, sum = -21
And listing out sums from part (a):
- When n = 1, sum = 4
- When n = 2, sum = 20
- When n = 3, sum = 56
- When n = 4, sum = 120
And listing out sums from part (b):
- When n = 1, sum = 1
- When n = 2, sum = 10
- When n = 3, sum = 35
- When n = 4, sum = 84
I found a pattern between the three sums:
- Part (c) sum = -3 = 1-4
- Part (c) sum = 6 = 10-4
- Part (c) sum = -10 = 10-20
- Part (c) sum = 15 = 35-20
- Part (c) sum = -21 = 56-35
I don’t know how to proceed from here. Any help is GREATLY appreciated! Please let me know if you need any clarification.
Kendra :)