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.
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
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Question
Help me with 6.10
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
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