4 (10 points) Encrypt the message STOP POLLUTION by translating the letters into

Question

4 (10 points) Encrypt the message STOP POLLUTION by translating the letters into numbers, applying the given encryption function, and then translating the numbers back into letters. a) f (p) (p 4) mod 26 5. (10 points) Prove that 1 1 2 2!+. .+n n! (n 1)! -1 whenever n is a positive integer. 6. (10 points) Prove that 2 2 (-7) 2 (-7)2 ..+2 (-7)n (1 -(-7) n+1)/4 Whenever n is a nonnegative integer. 7. (15 points)Find f (1), f (2), f (3), f (4), and f (5) if f (n) is defined recursively by f (0) 1 and for n 0, 1, a) f (n 1) f (n) 10 c) f (n 1) 2 (n) 5. f(n) e) f (n 1 2

Explanation / Answer

5.

nn!={(n+1)1}n!=(n+1)n!n!=(n+1)!n!

Therefore, when you add all n.n!,

When you sum up all things: 1.1!+2.2! ... n.n! = (n+1)!n! +n! - (n-1)! +...-1! = (n+1)!-1!

Hence proved

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