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# p6 edexcel rev. exe. qn 29. proof by induction watch

1. desperately need this solutions. panic time
2. (Original post by ruzaika)
desperately need this solutions. panic time
Post the question.
3. (Original post by Galois)
Post the question.
here goes

given that n elemnt of Z, Φ element real nos. and

M = (cosh²Φ cosh²Φ )
(-sinh²Φ -sinh²Φ )

use induction to prove that M^n = M.

by the way m is a matrix. in case a bunch of smilies appear after cosh² and -sinh², please take it as Φ.
4. Base step
The result is true for n = 1 because, by the rules of matrix powers, M^1 = M.

Inductive step
Suppose that the result is true for n = k. Then

M^(k + 1)
= M^k M . . . . . by the rules of matrix powers
= M^2 . . . . . by the inductive hypothesis
=
=
=
= M

So the result is also true for n = k + 1.

Conclusion
By induction, the result is true for all n >= 1.
5. (Original post by Jonny W)
Base step
The result is true for n = 1 because, by the rules of matrix powers, M^1 = M.

Inductive step
Suppose that the result is true for n = k. Then

M^(k + 1)
= M^k M . . . . . by the rules of matrix powers
= M^2 . . . . . by the inductive hypothesis
=
=
=
= M

So the result is also true for n = k + 1.

Conclusion
By induction, the result is true for all n >= 1.
am i grateful to you or what. makes loads of sense to me. the penny has dropped.

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