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Prove by induction that for all integers n>3:
3n > 9n
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Problem by BogusBoy
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Give a proof by induction to show that for every non-negative integer n:
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4
Additional Question: Could this induction be extended to all integers, not just negative ones? If so, how?
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Sum of Integers Proof
1 = 1 =
1 + 2 = 3 =
1 + 2 + 3 = 6 =
1 + 2 + 3 + 4 = 10 =
1 + 2 + 3 + 4 + 5 = 15 =
It appears from this that the sum of the first n positive integers is . Can you prove this by induction?
1(1 + 1)
2
1 + 2 = 3 =
2(2 + 1)
2
1 + 2 + 3 = 6 =
3(3 + 1)
2
1 + 2 + 3 + 4 = 10 =
4(4 + 1)
2
1 + 2 + 3 + 4 + 5 = 15 =
5(5 + 1)
2
n(n + 1)
2
Series Proof
Use a proof by induction to prove that the first n terms of the series
1
2
1
4
1
8
1
2n
2n - 1
2n
Eleven to the N
Prove by induction that for every integer n ≥ 1, 11n is one more than a multiple of ten.
Note: Proof by induction is not the simplest method of proof for this problem, so an alternate solution is provided as well.
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