Confused about this answer, michael spivak's calculus

In summary, the given sum is the sum of all of the odd numbers up to 2n. The solution involves rewriting the sum as the sum of all numbers up to 2n minus the sum of all even numbers up to 2n. The formula n*(n+1)/2 can be used to find the sum of all numbers up to n.
  • #1
rlpp
3
0

Homework Statement


Find the formula
Σ[n, i =1](2i - 1) = 1+3+5+...+(2n-1)


Homework Equations





The Attempt at a Solution


When I look at the back of the book , I got this solution which is

1, Sigma(2i - 1) = 1+3+5+...+(2n-1)

2, = 1+2+3+...+2n-2(1+..+n)

3, = ((2n)(2n+1)) /2 - n(n+1)

4, n^2

What i don't get is what happened from step 1 to 2 and 2 to 3?
It seems that there are hidden steps there that were not written down and therefore and I am really confused about this solution.
Thank you guys very much.
 
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  • #2
Your given sum is the sum of all of the odd numbers up to 2n. Step 2 is pretty easy. He just rewrote that as the sum of ALL of the numbers up to 2n minus the sum of all of the even numbers up to 2n. The trick you might be missing is that the sum of all of the numbers up to n, Sigma(i) for i=1 to n is given by a formula. It's n*(n+1)/2. There's at least a couple of ways to prove that. Do you know either one?
 
  • #3
Thank you very much, I am still a little confused but I will work with this formula. Spivak's calculus is so confusing lol.
 

1. What is "Confused about this answer, Michael Spivak's Calculus"?

"Confused about this answer, Michael Spivak's Calculus" is a common phrase used by students who are struggling to understand a particular concept or question in the textbook "Calculus" written by Michael Spivak. It is often used as a subject line in online forums or study groups to ask for help or clarification.

2. Who is Michael Spivak?

Michael Spivak is an American mathematician and writer. He is best known for his textbook "Calculus," which is considered a classic in the subject. Spivak has also written several other books on mathematics and is a professor at the University of Texas at Austin.

3. Why is "Calculus" by Michael Spivak considered a challenging textbook?

"Calculus" by Michael Spivak is considered a challenging textbook because it presents the subject in a rigorous and abstract manner. It requires a strong understanding of algebra and geometry, as well as the ability to think critically and solve complex problems. The book also includes advanced topics and proofs, making it a popular choice for advanced undergraduate courses.

4. Where can I find solutions to the exercises in "Calculus" by Michael Spivak?

Unfortunately, there are no official solutions provided for the exercises in "Calculus" by Michael Spivak. However, there are many online resources and study groups where students discuss and provide solutions to the exercises. It is also recommended to work through the problems with a tutor or study group to better understand the concepts.

5. How can I better understand "Calculus" by Michael Spivak?

To better understand "Calculus" by Michael Spivak, it is important to have a strong foundation in algebra and geometry. It is also helpful to work through the exercises and seek help from a tutor or study group when needed. Additionally, reading the book slowly and carefully, taking notes, and reviewing the material regularly can also improve understanding.

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