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3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.
Let a1 = 3 and an+1 = a + 5 2an for all n > 1 Prove that (an)nen converges and find limn7oo an:
i. (2nd Principle of Induction): Suppose that a1 = 2 and a2 = 4 and for n > 2, an = 5an-1 – 6an-2. Prove that for all n e N, an = 2". (This is easy. Show precisely where you need the 2nd Principle.)
Use the Principle of mathematical induction to prove
2. Use the Principle of Mathematical Induction to prove: Lemma. Let n E N with n > 2, and let al, aa-.., an E Z all be nonzero. If gcd(ai ,aj) = 1 for all i fj, then gcd(aia2an-1,an)1. 1, a2,, an
Let ne Nj. Prove that n < 2(6(n)).
8. Let f:D → R and let c be an accumulation point of D. Suppose that lim - cf(x) > 1. Prove that there exists a deleted neighborhood U of c such that f(x) > 1 for all 3 € Un D.
Let U ? Rmxn. Prove that if UTI-In, then n < m.
PLEASE HELP WITH PROOF!!
8. Let an > 0 for all n in 1. Show that if an converges, then Ĉ vanın converges. [Hint: Expand [van - (1/n)]2.) N =
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.
(1 point) The matrix A = 1-6 1 8 k] 4 has two distinct real eigenvalues if and only if k > 24.5