Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Tens

Stage: 5 Challenge Level:

(1) Show that for all odd natural numbers $n$ both $9^n + 1^n$ and $7^n + 3^n$ are multiples of $10$.

(2) Show that for all even natural numbers $n$ both $8^n - 2^n$ and $6^n - 4^n$ are multiples of $10$.

(3) Find and prove similar results to (1) for even powers.

(4) Find and prove similar results to (2) for odd powers.

NOTES AND BACKGROUND
There are eight results to be proved. You are invited to spot patterns, to make conjectures and then to prove your conjectures.

The two principal methods of proof involve ideas central to mathematics courses in the last two years of school. Finding both methods is one of the challenges here.

If you are proficient you won't need to prove all 8 cases once you have demonstrated that you can prove one, but if you do not find the proofs easy then it will be good practice to prove the results in all the cases.

Modulus arithmetic. Fibonacci sequence. Mathematical induction. Trigonometric functions and graphs. Factors and multiples. Binomial Theorem. Periodicity. Making and proving conjectures. Mathematical reasoning & proof. Index notation/Indices.

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Stage: 5 Challenge Level: