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dmitriy555 [2]
3 years ago
15

Using mathematical induction prove whether or not the following statement is true for all positive integers n, or show why it is

false.
2+6+10+ ... +4n-2=2n^2
Mathematics
1 answer:
aniked [119]3 years ago
5 0

Base case: For n=1, the left side is 2 and the right is 2\cdot1^2=2, so the base case holds.

Induction hypothesis: Assume the statement is true for n=k, that is

2+6+10+\cdots+4k-2=2k^2

We want to show that this implies truth for n=k+1, that

2+6+10+\cdots+4k-2+4(k+1)-2=2(k+1)^2

The first k terms on the left reduce according to the assumption above, and we can simplify the k+1-th term a bit:

\underbrace{2+6+10+\cdots+4k-2}_{2k^2}+4k+2

2k^2+4k+2=2(k^2+2k+1)=2(k+1)^2

so the statement is true for all n\in\mathbb N.

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