Induction: 1+2+2^2+2^3+ +2^n-1=2^n-1

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In summary: You get 2^(k+1)!In summary, You are trying to prove that if, for some k, 2^0+ 2^1+ \cdot\cdot\cdot+ 2^{k-1}= 2^k-1 then 2^0+ 2^1+ \cdot\cdot\cdot+ 2^{k-1}+ 2^k= 2^{k+1}-1.
  • #1
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1. Homework Statement [/b]
Use mathematical induction to prove the following statements are true for all integers n≥1
1+2+2^2+2^3+...+2^n-1=2^n-1



Attempt at a solution:


For n=1

1+2^(1-1)=2^1-1
1=1
∴ it is true.


Let Sn=Sk
If it is true for k, it must also be true for k+1

1+2+2^2+2^3+...+2^k+2^k+1-1=(2^k+1)-1

This is the part I have a slight confusion at, I keep getting something left over and the sides don't equal each other :(
 
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  • #2
Daaniyaal said:
1. Homework Statement [/b]
Use mathematical induction to prove the following statements are true for all integers n≥1
1+2+2^2+2^3+...+2^n-1=2^n-1



Attempt at a solution:


For n=1

1+2^(1-1)=2^1-1
1=1
∴ it is true.


Let Sn=Sk
If it is true for k, it must also be true for k+1

1+2+2^2+2^3+...+2^k+2^k+1-1=(2^k+1)-1

This is the part I have a slight confusion at, I keep getting something left over and the sides don't equal each other :(

Use brackets: you have written
1+2+2^2+2^3+...+2^n-1=2^n-1 , which is plainly impossible, because it would require that we have 1+2+2^2+2^3+... = 0 (in order to have 2^n-1=2^n-1 on both sides). If you mean 2^(n-1), then that is what you need to say.

RGV
 
  • #3
You want to prove that if, for some k, [itex]2^0+ 2^1+ \cdot\cdot\cdot+ 2^{k-1}= 2^k-1[/itex] then [itex]2^0+ 2^1+ \cdot\cdot\cdot+ 2^{k-1}+ 2^k= 2^{k+1}-1[/itex].
(What you wrote, 1+ 2^1+ ...+ 2^n-1= 2^n-1 is, as Ray Vickson said, clearly impossible because you have "2^n- 1" on both sides but with additional positive terms on the left. Of course, you meant 2^(n-1) on the left and (2^n)- 1 on the right. Those are very different and you can't ask people to guess what you mean.)

To go from k to k+1, obviously you are adding [itex]2^k[/itex] to the sum on the left so do it on the right:
[tex]2^0+ 2^1+ \cdot\cdot\cdot+ 2^{k-1}+ 2^k= (2^k- 1)+ 2^k[/tex]

What do you get when you combine those two "[itex]2^k[/itex]" terms on the right?
 
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  • #4
I figured it out, thanks!
 

1. How can you prove that 1+2+2^2+2^3+...+2^n-1=2^n-1?

The proof for this equation involves using mathematical induction, which is a method of mathematical proof that is used to prove statements about all natural numbers. The basis step involves proving that the statement is true for the first natural number (in this case, n=1), and the induction step involves showing that if the statement is true for one natural number, it is also true for the next natural number (in this case, n+1). By completing both steps, we can prove that the equation is true for all natural numbers.

2. What is the purpose of using mathematical induction in this equation?

The purpose of using mathematical induction is to prove that a statement or equation is true for all natural numbers. In this case, we are using induction to prove that the equation 1+2+2^2+2^3+...+2^n-1=2^n-1 is true for all values of n. Induction allows us to make a general statement about a series of numbers based on a few specific examples.

3. What is the significance of this equation in mathematics?

This equation is significant because it shows the relationship between the sum of a geometric series and its final term. It is a fundamental concept in mathematics and is used in a variety of fields, including calculus, number theory, and computer science. Additionally, it demonstrates the power and usefulness of mathematical induction as a proof technique.

4. Can this equation be applied to other series?

Yes, this equation can be applied to other series that follow a similar pattern. This equation is a specific case of the more general formula for the sum of a geometric series, which is given by S_n = a(1-r^n)/(1-r), where a is the first term, r is the common ratio, and n is the number of terms. As long as a series follows this pattern, we can apply this equation to find the sum.

5. How is this equation used in real-life applications?

This equation is used in a variety of real-life applications, such as calculating compound interest, determining the growth rate of populations, and analyzing the performance of algorithms. It can also be used in fields such as physics and engineering to model exponential growth and decay. Overall, this equation has many practical uses and is a fundamental concept in many areas of science and mathematics.

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