1. Prove that 123 + 234 +...+ n(n+1)(n+2) = n(n + 1)(n + 2)(n + 3)/4 by indu

Q: 1. Prove that 1*2*3 + 2*3*4 +...+ n(n+1)(n+2) = n(n + 1)(n + 2)(n + 3)/4 by indu

1. Base Case : Let n = 1. Then the left-hand side of (*) is 1

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1. Prove that 1*2*3 + 2*3*4 +...+ n(n+1)(n+2) = n(n + 1)(n + 2)(n + 3)/4 by induction on n.

2. Suppose there are n people in a group, and each person knows a secret that no other member of the group knows. Whenever two members of the group talk on the phone, they tell each other all of the secrets that they know. Let G(n) be the minimum number of phone calls required so that all n members the group know every secret.

(a) (4 pts) Find G(1), G(2), G(3), and G(4). Justify your answers.

(b) (7pts)ProvethatG(n)?2n?4 for n?4 by induction on n.

Explanation / Answer

Base Case : Let n = 1.

Then the left-hand side of (*) is 1

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1. Prove that 1*2*3 + 2*3*4 +...+ n(n+1)(n+2) = n(n + 1)(n + 2)(n + 3)/4 by indu

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1. Prove that 123 + 234 +...+ n(n+1)(n+2) = n(n + 1)(n + 2)(n + 3)/4 by indu