3. Define Cauchy sequence and prove that the sequence { } is a Cauchy sequence in C[1,3]. nt+5

Define a sequence by an+1 = -an?, with -1 < ao < 0. Prove that {an} www is increasing and bounded above.

suppose a(n + 2) = a(n) + a(n + 1) with a(1) = 1 and a(2) = 1 %3D SELE %3D A) the sequence is an geometric sequence B) the initial terms of the sequence are: {1,1,2, 3, 5, ..} C) the sequence is an arithmetic sequence D) the closed form is a(n) = ±()" -()"

1 and 2an + 2" for all integers n, n > 1. Use induction to prove that 2. (4 points) Suppose {an} is a sequence recursively defined by a1 = ап+1 || an = n2"-' for all positive integers n.

2. Consider the sequence {an} defined by ai = 3, a2 = 7, and an integer n 3. Зап-1 — 2ат-2 for every (а) Compute аз, ад, аnd as- (b) Formulate a conjecture about an (i.e. express an in terms of n) (c) Use strong mathematical induction to prove your conjecture.

4. Suppose that (an) is a sequence with an > 0 for all n. Define a sequence (bn) by + An a₁ + a₂ + n Show that bn diverges. bn =