All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(a)

Let P(n) be the statement

When n=1, LHS

RHS

Since LHS = RHS, P(1) is true.

Assume P(k) is true for some k,

Want to show P(k+1) is true,

LHS

= RHS

Since P(1) is true, P(k) is true P(k+1) is true, hence, by Mathematical Induction, P(n) is true for all

(b)

(i)

(ii)

(iii)

Least n = 499