Multiples of Pi/4
Pro Problems > Math > Logic > Proofs > Proof by InductionMultiples of Pi/4
Prove by induction that for any non-negative integer n,
tan ((4n + 1)p
4
Additional Question: Could this induction be extended to all integers, not just negative ones? If so, how?
Note: Without rigorous proof, we can see that this is true, since the angles which match the equation are all in the first and third quadrant, and all have reference triangles ofp
4
Presentation mode
Problem by BogusBoy
Solution
In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.
Assign this problem
Click here to assign this problem to your students.Similar Problems
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
Geometric Sum Proof
Give a proof by induction to show that for every non-negative integer n:
20 + 21 + 22 + ... + 2n = 2n + 1 - 1
Square Sum Proof
Prove by induction that the sum of the first n positive perfect squares is:
n(n + 1)(2n + 1)
6
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.
Product and Power
Prove by induction that for all integers n>3:
3n > 9n
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
Blogs on This Site
Like us on Facebook to get updates about new resources