(1) Prove that for Res > 3 (s)S(s - 2)- n-1 where σ2(n)-2.1n d2 is the...
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,
5. Prove that U(2") (n > 3) is not cyclic.
Prove each problem, prove by induction 3) Statementn-1 5 25(2m-1) forn2 1 4 Statement Suppose: bo1 . b,-2b-1 + 1 for t 1 en fort >
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
2. Assume X is a random variable following from N(μ, σ2), where σ > 0. (a) Write down the pdf of X (b) Compute E(X2). (b) Define YFind the distribution of Y.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
Prove that there are no natural number solutions to the equation where x, y ≥ 2 ... (See Picture Below) Prove that there are no natural number solutions to the equation where X, Y > 2. x2 - y2 = 1.
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.