Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Part 2: Applying the Formula for the Sum of the First n integers as shown in 5.2. Make up an original problem similar to example 5.2.2
Evaluate
a. Evaluate 2+ 4 + 6 + ... + 200.
b. Evaluate 1+ 2 + 3+4+ ... + 300.
c. 5+7+9+11+...+159.
b. Evaluate 1+ 2 + 3+4+ ... + 300.
c.
![### Theorem 5.2.1: Sum of the First n Integers
For every integer \( n \geq 1 \),
\[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \]
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### Definition
If a sum with a variable number of terms is shown to equal an expression that does not contain either an ellipsis or a summation symbol, we say that the sum is written in **closed form**.
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This content explains a fundamental theorem and associated definition in mathematics, particularly on the topic of summing the first \( n \) integers. The theorem provides a formula to find the sum of the first \( n \) natural numbers efficiently, which is represented in a closed form as \( \frac{n(n+1)}{2} \).](https://content.bartleby.com/qna-images/question/e2b4e390-596f-48e2-849d-32f83635f844/0f1f0c5b-10f0-4a97-91ad-aed60c47d1b0/wz2n8d7.png)
Transcribed Image Text:### Theorem 5.2.1: Sum of the First n Integers
For every integer \( n \geq 1 \),
\[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \]
---
### Definition
If a sum with a variable number of terms is shown to equal an expression that does not contain either an ellipsis or a summation symbol, we say that the sum is written in **closed form**.
---
This content explains a fundamental theorem and associated definition in mathematics, particularly on the topic of summing the first \( n \) integers. The theorem provides a formula to find the sum of the first \( n \) natural numbers efficiently, which is represented in a closed form as \( \frac{n(n+1)}{2} \).
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- Complete the conjecture based on the pattern you observe in the examples below. 5. Conjecture: The sum of any two odd numbers is 1+1=2 1+5=6 7+9=16 9 + 11 = 20 13 +21 = 34 101 + 103 = 204 6. Conjecture: The product of any two odd numbers is 1x1 = 1 1x3 = 3 3x7 = 21 1.2 Patterns and Conjectures 9x1199 13x11 = 143 101x103 = 10,403 7. Conjecture: The product of a number (x - 1) and the number (x + 1) is always equal to 3x5 = 4² - 1 2 7x9 = 8²-1 4x6 = 5² - 1 6x8 = 7² - 1 5x7 = 6²-1 8x10 = 9²-1 Page 1 of 1arrow_forward1. Consider a 4 digit number abcd and show that abcd−(a+b+c+d) is alwaysdivisible by 9. (Ex 1234 − 10 = 1224 = 9 · 136). 2. (A) How many different three-letter initials can a person have?(B) What if the initials are not repeated?(C) What if only consecutive initials are not repeated? ( please just answer part c on this one.. A and B are just so you know the entire question)arrow_forwardfind a formula for the sum 2+4+6+..2n when n is a natural number and prove that your formula worksarrow_forward
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