Prove that for all natural numbers n, 1 +2 + 3 + .. + n = , using mathematical induction. Hint: Follow these steps below to prove the problem: i. Show that it is true for n = 1. ii. Assume it is true for n = k. iii. Prove it is true for k+ 1. BIU EE s° S, πο 2.
Prove that for all natural numbers n, 1 +2 + 3 + .. + n = , using mathematical induction. Hint: Follow these steps below to prove the problem: i. Show that it is true for n = 1. ii. Assume it is true for n = k. iii. Prove it is true for k+ 1. BIU EE s° S, πο 2.
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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![### Mathematical Induction: Proving the Sum Formula
#### Problem Statement:
Show that for all natural numbers \( n \), the formula for the sum of the first \( n \) natural numbers is given by:
\[
1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2}
\]
using mathematical induction.
#### Hint:
Follow these steps below to prove the problem:
i. **Base Case**: Show that it is true for \( n = 1 \).
ii. **Inductive Hypothesis**: Assume it is true for \( n = k \).
iii. **Inductive Step**: Prove it is true for \( k + 1 \).
#### Text Box for Work:
(Include a space to write or type the proof steps)
This guidance helps in using mathematical induction to establish the given formula's validity for all natural numbers. Follow each step carefully, ensuring clarity and completeness in the proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2cad0f5b-530d-4248-ba5a-31b2999fa76b%2F3fcccd0c-1299-4e96-b27e-8b87f6cefb1a%2Fchp01pd_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Induction: Proving the Sum Formula
#### Problem Statement:
Show that for all natural numbers \( n \), the formula for the sum of the first \( n \) natural numbers is given by:
\[
1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2}
\]
using mathematical induction.
#### Hint:
Follow these steps below to prove the problem:
i. **Base Case**: Show that it is true for \( n = 1 \).
ii. **Inductive Hypothesis**: Assume it is true for \( n = k \).
iii. **Inductive Step**: Prove it is true for \( k + 1 \).
#### Text Box for Work:
(Include a space to write or type the proof steps)
This guidance helps in using mathematical induction to establish the given formula's validity for all natural numbers. Follow each step carefully, ensuring clarity and completeness in the proof.
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