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Work out 2^n+3^n for some different odd values of n.
What do you notice?

n 2^n Pattern 3^n Pattern
1 2 x4 3 x9
3 8 x4 27 x9
5 32 x4 243 x9
7 128 x4 2187 x9

2^n always ends in 2 or 8 because you multiply by four every time you increase n by 2. This is because you’re in the powers 2^n so 2x2 = 4 so you square the number. I can prove this works by showing that it also works on the number 3. You times the 3^n by 9 every time you increase n by 2. This is because you do 3x3 = 9. If you add 2^n+3^n(and n is odd) it always equals 5 because each one alternates between 2 and 8 for〖 2〗^n, and 3 and 7 for 2^n. So if it goes 2+3 then 8+7 they both end in 5 so therefore it always ends in 5.