Induction (Geometric Series): need help on this problem, attached is a sample problem to help with answer Using induction, prove the geometric series formula:Eapi-11- rn= a11-ri=1(You may find it helpful to see the practice problem on induction to see what is required ofyour proof). Using induction, show that n! < n" for all n > 1. (You may find it helpful to see the practiceproblem on induction to see what is required of your proof).Solution:We use induction on a sequence of statements:• 2! < 22 (statement 2)• 3! < 33 (statement 3)• n! < n" (statement n)Base Case (statement 2):2! = 2 * 1 = 2 < 4 = 2²so that 2! < 22Inductive Step: Assume statement n is true, that is n! < n". Then:(n +1)! = n! * (n + 1)< п" * (п+1)< (n +1)" * (п + 1)= (n +1)"+1Where we use statement n in the second line and the fact that n" < (n +1)" in the third (since nis positive). This statement shows that (n +1)! < (n + 1)"n+1.By induction, n! < n" for all n > 1.

We have to prove the geometric series formula ∑i=1na1ri-1=a11-rn1-r. Solution: We will prove the…

Answered: Using induction, prove the geometric…

Using induction, prove the geometric series formula: n 1- rn = a1 1 - r i-1 i=1 (You may find it helpful to see the practice problem on induction to see what is required of your proof).

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

Induction (Geometric Series): need help on this problem, attached is a sample problem to help with answer

Using induction, prove the geometric series formula:
Eapi-1
1- rn
= a1
1-r
i=1
(You may find it helpful to see the practice problem on induction to see what is required of
your proof).

Using induction, show that n! < n" for all n > 1. (You may find it helpful to see the practice
problem on induction to see what is required of your proof).
Solution:
We use induction on a sequence of statements:
• 2! < 22 (statement 2)
• 3! < 33 (statement 3)
• n! < n" (statement n)
Base Case (statement 2):
2! = 2 * 1 = 2 < 4 = 2²
so that 2! < 22
Inductive Step: Assume statement n is true, that is n! < n". Then:
(n +1)! = n! * (n + 1)
< п" * (п+1)
< (n +1)" * (п + 1)
= (n +1)"+1
Where we use statement n in the second line and the fact that n" < (n +1)" in the third (since n
is positive). This statement shows that (n +1)! < (n + 1)"n+1.
By induction, n! < n" for all n > 1.

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