n (7) Vn € N, Σi!i = (n + 1)! — 1. i=1
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
Prove by induction.
![The given mathematical formula expresses a summation identity for natural numbers:
\[
(7) \quad \forall n \in \mathbb{N}, \, \sum_{i=1}^{n} i \cdot i = (n+1)! - 1.
\]
This equation states that for any natural number \( n \), the sum of the squares of the numbers from 1 to \( n \) multiplied by their index \( i \) is equal to the factorial of \( n+1 \) minus 1.
### Explanation of Terms:
- **\(\forall n \in \mathbb{N}\)**: This denotes "for all \( n \)" where \( n \) is a member of the set of natural numbers.
- **\(\sum_{i=1}^{n} i \cdot i\)**: This is the summation notation, indicating that you sum over the index \( i \) from 1 to \( n \) where each term in the sum is the product of \( i \) and itself.
- **\((n+1)!\)**: This is the factorial of \( n+1 \), which means multiplying all positive integers up to \( n+1 \).
- **\(-1\)**: This indicates that 1 is subtracted from the result of the factorial.
This identity showcases a relationship between summation and factorials, providing insight into combinatorial mathematics and algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F868daa41-93eb-4d05-be30-2d6217bf0561%2F1f1f6b2d-77cc-435b-bb65-bb7f966d2a91%2Fdwm46f_processed.png&w=3840&q=75)
Transcribed Image Text:The given mathematical formula expresses a summation identity for natural numbers:
\[
(7) \quad \forall n \in \mathbb{N}, \, \sum_{i=1}^{n} i \cdot i = (n+1)! - 1.
\]
This equation states that for any natural number \( n \), the sum of the squares of the numbers from 1 to \( n \) multiplied by their index \( i \) is equal to the factorial of \( n+1 \) minus 1.
### Explanation of Terms:
- **\(\forall n \in \mathbb{N}\)**: This denotes "for all \( n \)" where \( n \) is a member of the set of natural numbers.
- **\(\sum_{i=1}^{n} i \cdot i\)**: This is the summation notation, indicating that you sum over the index \( i \) from 1 to \( n \) where each term in the sum is the product of \( i \) and itself.
- **\((n+1)!\)**: This is the factorial of \( n+1 \), which means multiplying all positive integers up to \( n+1 \).
- **\(-1\)**: This indicates that 1 is subtracted from the result of the factorial.
This identity showcases a relationship between summation and factorials, providing insight into combinatorial mathematics and algebra.
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