8. (14 points) Let dj = 1, a2 = 4, and an = 2an-1 - An-2+2 for n > 3. Prove that an = n2 for all all natural numbers n.
Let an be the recurrence defined by: ao = 4.4 = 7, and for all n 2, an-2an-1 + 5an-2. Using constructive induction, find integer constants A and B such that for all n 2 0, an S AB". Try to make B as small as possible. Let an be the recurrence defined by: ao = 4.4 = 7, and for all n 2, an-2an-1 + 5an-2. Using constructive induction, find integer constants A and B such that for all...
20. (4 pts) Consider the following recurrence. an = 2an-1 + 2an-2 ao = 0 Q1 = 2V3 For what values of a and B is the following expression a solution of that recurrence? a;=a (1+ v3)*+B (1 - v3)' a = -1 and B 1 O a = = { and B :- O a = 2 and B = -2 the four other possible answers are incorrect O a = 1 and B = -1
Let a1 = 3 and an+1 = a + 5 2an for all n > 1 Prove that (an)nen converges and find limn7oo an:
Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and an a/(n-1) for n22. Prove using strong induction that an for any n2 1 2. Consider a sequence {an} defined by recurrence: a1 = 1, a2-1 and an-2an-1 +an-2 for n 2 3. Prove using strong induction that an K 3" for any n21 Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and...
Prove the following: 1+4+7+...+(3n – 2) n(3n-1) 2
-1) Prove that 12+22 + ... + n2 = n(n + 1) (2n + 1) 6 -1) Prove that 12+22 + ... + n2 = n(n + 1) (2n + 1) 6
Problem 8: (i) Use the Principle of Mathematical Induction to prove that 2n+1(-1)" + 1 1 – 2 + 22 – 23 + ... + (-1)22" = for all positive integers n. (ii) Use the Principle of Mathematical Induction to prove that np > n2 + 3 for all n > 2.
Please Prove. Prove 2 n > n2 by induction using a basis > 4: Basis: n 5 32> 25 Assume: Prove: