A sequence a1, a2, a3, ... is defined as follows: a1 = 3, and ak = 4ak-1 + 2 for all integers k ≥ 2. Supposing that a5 = 44.3 + 43.2 + 42.2 +4-2+2, find a similar numerical expression for a6 by substituting the right-hand side of this equation in place of as in the equation a6 = 4.25 +2. O 45.3+44.2 + 43.2+42.2+4.2+2 O 44.3+43.2+42.2+4.2+(2+2) O 44.3+43-2 +4²-2+4.2+4.2 O 45 12+44-8+43.8+42-2+4.8+2 Use mathematical induction to prove that for all integers n ≥ 1, 4+8+12+...+ 4n = 2n² + 2n. Which steps would be necessary? O a. Show that P(1) is true b. Show that for all integers k≥ 1, if P(k) is true then P (k+1) is true. O a. Show that P(even) is true b. Show that Plodd) is true. O a. Show that P(1) is true b. Create a contradiction for all integers k≥ 1, if P(k) is false then P (k+1) is false. O a. Show that P(1) is true b. Show that P(2) is true. c. Show that P(3) is true. d. Show that P(4n) is true, for all n.
A sequence a1, a2, a3, ... is defined as follows: a1 = 3, and ak = 4ak-1 + 2 for all integers k ≥ 2. Supposing that a5 = 44.3 + 43.2 + 42.2 +4-2+2, find a similar numerical expression for a6 by substituting the right-hand side of this equation in place of as in the equation a6 = 4.25 +2. O 45.3+44.2 + 43.2+42.2+4.2+2 O 44.3+43.2+42.2+4.2+(2+2) O 44.3+43-2 +4²-2+4.2+4.2 O 45 12+44-8+43.8+42-2+4.8+2 Use mathematical induction to prove that for all integers n ≥ 1, 4+8+12+...+ 4n = 2n² + 2n. Which steps would be necessary? O a. Show that P(1) is true b. Show that for all integers k≥ 1, if P(k) is true then P (k+1) is true. O a. Show that P(even) is true b. Show that Plodd) is true. O a. Show that P(1) is true b. Create a contradiction for all integers k≥ 1, if P(k) is false then P (k+1) is false. O a. Show that P(1) is true b. Show that P(2) is true. c. Show that P(3) is true. d. Show that P(4n) is true, for all n.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A sequence a1, a2, a3, ... is defined as follows:
a1 = 3, and ak = 4ak-1 + 2 for all integers k ≥ 2.
Supposing that a5 = 44.3 + 43.2 + 42.2 +4-2+2, find a similar numerical expression for a6 by
substituting the right-hand side of this equation in place of as in the equation a6 = 4.25 +2.
O 45.3+44.2 + 43.2+42.2+4.2+2
O
44.3+43.2+42.2+4.2+(2+2)
O 44.3+43-2 +4²-2+4.2+4.2
O 45 12+44-8+43.8+42-2+4.8+2
Transcribed Image Text:Use mathematical induction to prove that for all integers n ≥ 1,
4+8+12+...+ 4n = 2n² + 2n.
Which steps would be necessary?
O a. Show that P(1) is true
b. Show that for all integers k≥ 1, if P(k) is true then P (k+1) is true.
O a. Show that P(even) is true
b. Show that Plodd) is true.
O a. Show that P(1) is true
b. Create a contradiction for all integers k≥ 1, if P(k) is false then P (k+1) is false.
O a. Show that P(1) is true
b. Show that P(2) is true.
c. Show that P(3) is true.
d. Show that P(4n) is true, for all n.
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