Homework Answers
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38...
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.
8. (14 points) Let dj = 1, a2 = 4, and an = 2an-1 - An-2+2...
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.
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an...
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an...
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.
PLEASE HELP WITH PROOF!! 8. Let an > 0 for all n in 1. Show that...
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 =
5. Prove that U(2") (n > 3) is not cyclic.
5. Prove that U(2") (n > 3) is not cyclic.
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4)...
.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
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show...
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show that T is not diagonalizable.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
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.
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.
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.
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 =
5. Prove that U(2") (n > 3) is not cyclic.
.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
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show that T is not diagonalizable.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
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