Q2. Induction. For each integer n > 1, define the sum S(n) as follows: 1 1 1 1 1 S(n) = (2i — 1)(2і + 1) 3.5 5- 7 (2n – 1)(2n + 1)* n=1 1 2 1 1 For example S(1) =E and S(2) = %3D %3D (2i – 1)(2i +1) 1.3 (2i – 1)(2i + 1) 1-3 3-5 n=1 i=1 - i. Compute each of the following values of S(n) and write your final answer as a fraction in lowest terms. S(3) = S(4) = S(5) = - ii. Do you see a general pattern for the simplified value of S(n)? Write your guess to complete the proposition P(n) below. For each integer n > 1, let P(n) be the proposition defined as follows: 1 P(n): S(n) = E (2i – 1)(2i + 1) i=1 – iii. Prove that the proposition P(n) is true for all n 21 using a Proof by induction.

Q2. Induction. For each integer n > 1, define the sum S(n) as follows: 1 1 1 1 1 S(n) = (2i — 1)(2і + 1) 3.5 5- 7 (2n – 1)(2n + 1)* n=1 1 2 1 1 For example S(1) =E and S(2) = %3D %3D (2i – 1)(2i +1) 1.3 (2i – 1)(2i + 1) 1-3 3-5 n=1 i=1 - i. Compute each of the following values of S(n) and write your final answer as a fraction in lowest terms. S(3) = S(4) = S(5) = - ii. Do you see a general pattern for the simplified value of S(n)? Write your guess to complete the proposition P(n) below. For each integer n > 1, let P(n) be the proposition defined as follows: 1 P(n): S(n) = E (2i – 1)(2i + 1) i=1 – iii. Prove that the proposition P(n) is true for all n 21 using a Proof by induction.

Name: Q2.Induction. For each integer n > 1, define the sum S(n) as follows:
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Description: Q2.Induction. For each integer n > 1, define the sum S(n) as follows:1111S(n) =3.5+5.71.+..+(2i — 1)(2і + 1)3(2n – 1)(2n + 1)n=11111.For example S(1) =L 2i – 1)(2i +1)and S(2) =>%3D%3D1.3(2i – 1)(2i + 1)1-33-5n=1i=1- i. Compute each of the following

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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Q2.
Induction. For each integer n > 1, define the sum S(n) as follows:
1
1
1
1
S(n) =
3.5+5.7
1.
+..+
(2i — 1)(2і + 1)
3
(2n – 1)(2n + 1)
n=1
1
1
1
1.
For example S(1) =L 2i – 1)(2i +1)
and S(2) =>
%3D
%3D
1.3
(2i – 1)(2i + 1)
1-3
3-5
n=1
i=1
- i. Compute each of the following values of S(n) and write your final answer as a fraction in
lowest terms.
S(3) =
S(4) =
S(5) =
%3D
- ii. Do you see a general pattern for the simplified value of S(n)? Write your guess to complete
the proposition P(n) below.
For each integer n > 1, let P(n) be the proposition defined as follows:
n
1
P(n) : S(n) = E
(2i – 1)(2i + 1)
i=1
iii. Prove that the proposition P(n) is true for all n > 1 using a Proof by induction.
You must clearly state your Induction Hypothesis and indicate when it is used during the proof of your Induction Step.
As usual you must declare what each variable in your solution represents and make it clear whether each step of your proof is
an assumption, something you are about to prove, or something that follows from a previous step or definition, etc. Attach
additional paper if necessary

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