6.9. Prove that 1.3+2 6.10. Let r #1 be a real number. Use induction to prove that a + ar + ar? + ...+ ar- = for every positive integer n. Prove that I+ for every positive integer n.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

Help me with 6.10

SECTION 6.1 EXERCISES
6.1. Which of the sets are well-ordered?
|
10 GAGLA
(b) S={-2, – 1, 0, 1, 2} mobans
nob sr
p is a prime} = {2, 3, 5, 7, 11, 13, 17, . }.
||
well-ordered.
2 Prove that every nonempty set of negative integers has a largest element.
..
%3D
(1) by mathematical induction and s
(2) by adding 1 + 3 + 5 +
· .
·+ (2n – 1) and (2n – 1) + (2n – 3) + · ·· +1.
|
|
65. Use mathematical induction to
prove that
<1+5+9+...+ (4n – 3) = 2n2 –n
for every positive integer n.
66 (a) We have seen that 12 + 2² + ·. .+n² is the number of squares in an n × n square composed of
n² 1 × 1 squares. What does 13 + 2³ + 3³ +. +n³ repreșent geometrically?
12
fn an (b) Use máthematical induction to prove that 13 + 23 + 33+...+n =
n²(n + 1)²
for every positive
Lorborsk
integer n.
4.
I 12 CO
6.7. Find another formula suggested by Exercises 6.4 and 6.5 and verify your formula by mathematical
induction.
6.8. Find a formula for 1+4 +7+• +(3n – 2) for positive integers n and then verify your formula by
mathematical induction.
6.9. Prove that 1.3+2.4+3.5+ . ..+n(n+ 2) = "(n+1)(2n+7) for every positive integer n.
9.
= a= for every
6.10. Let r + be a real number. Use induction to prove that a + ar + ar²+.+ ar"-1
positive integer n.
...
6.11. Prove that +as +…+
1.
(n+2)(n+3)
GEGL
for every positive integer n.
6.12, Consider the open sentence P(n): 9 + 13 +. . + (4n + 5) = 4n²+14n+1 w
where n e N.
...
%3D
2.
ule (a) Verify the implication P(k) = P(k + 1) for an arbitrary positive integer k.
(b) Is Vn e N, P(n) true?
llupitedmy
8.0 moos
Transcribed Image Text:SECTION 6.1 EXERCISES 6.1. Which of the sets are well-ordered? | 10 GAGLA (b) S={-2, – 1, 0, 1, 2} mobans nob sr p is a prime} = {2, 3, 5, 7, 11, 13, 17, . }. || well-ordered. 2 Prove that every nonempty set of negative integers has a largest element. .. %3D (1) by mathematical induction and s (2) by adding 1 + 3 + 5 + · . ·+ (2n – 1) and (2n – 1) + (2n – 3) + · ·· +1. | | 65. Use mathematical induction to prove that <1+5+9+...+ (4n – 3) = 2n2 –n for every positive integer n. 66 (a) We have seen that 12 + 2² + ·. .+n² is the number of squares in an n × n square composed of n² 1 × 1 squares. What does 13 + 2³ + 3³ +. +n³ repreșent geometrically? 12 fn an (b) Use máthematical induction to prove that 13 + 23 + 33+...+n = n²(n + 1)² for every positive Lorborsk integer n. 4. I 12 CO 6.7. Find another formula suggested by Exercises 6.4 and 6.5 and verify your formula by mathematical induction. 6.8. Find a formula for 1+4 +7+• +(3n – 2) for positive integers n and then verify your formula by mathematical induction. 6.9. Prove that 1.3+2.4+3.5+ . ..+n(n+ 2) = "(n+1)(2n+7) for every positive integer n. 9. = a= for every 6.10. Let r + be a real number. Use induction to prove that a + ar + ar²+.+ ar"-1 positive integer n. ... 6.11. Prove that +as +…+ 1. (n+2)(n+3) GEGL for every positive integer n. 6.12, Consider the open sentence P(n): 9 + 13 +. . + (4n + 5) = 4n²+14n+1 w where n e N. ... %3D 2. ule (a) Verify the implication P(k) = P(k + 1) for an arbitrary positive integer k. (b) Is Vn e N, P(n) true? llupitedmy 8.0 moos
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,