Proving Rational Sum of n Rational Numbers

[willbland]

Homework Statement

Prove via Mathematical Induction that, The sum of n rational numbers is rational.

Homework Equations

The Attempt at a Solution

Let N = 1
The sum of one rational number is the number it's self, which is a rational number.

Assume when n = n, the sum of n numbers is rational

proof:

Here's why my problem lies, i don't know what equation this is , or how i would make an equation to sub (n + 1) into ? I'm looking for any guidance that i can get, all help is apreciated..

Thank's alot,
-Will

 

[Mark44]

Homework Statement

Prove via Mathematical Induction that, The sum of n rational numbers is rational.

Homework Equations

The Attempt at a Solution

Let N = 1
The sum of one rational number is the number it's self, which is a rational number.

I would start with n = 2, and show that a/b + c/d is a rational number.

Assume when n = n, the sum of n numbers is rational

Assume that when n = k, r₁ + r₂ + ... + r_k is a rational number.

proof:

Here's why my problem lies, i don't know what equation this is , or how i would make an equation to sub (n + 1) into ? I'm looking for any guidance that i can get, all help is apreciated..

What is the statement you need to prove when n = k + 1?

 

[willbland]

Thank's a lot for your help Mark, you really steered me in the right direction here.

I proved that when n=2 the sum was rational using a/b + c/d etc etc etc, but after i assume that when n=n , for all natural numbers the sum is rational, isn't that as far as you'd think i would have to go? I mean there's no domino effect needed for this proof , just that the sum of n natural numbers is rational, or , do i have to do the n + 1 thing and prove for every n ? but i don't understand how i would go about doing that, :(

 

[Office_Shredder]

assume that when n=n

If you're ever in a situation where n is not equal to n, induction will be the least of your worries.

It doesn't sound like you completely understand what induction is supposed to be saying. You have to do two things: Prove the statement is true when n=2, which it sounds like you did, and prove that if the statement is true when n=k, it's true when n=k+1

For example, can you prove that the sum of three rational numbers is rational, using the fact that the sum of two rational numbers is rational? Can you prove that the sum of four rational numbers is rational, using the fact that the sum of three rational numbers is rational? Try to generalize this this for any k

 

[HallsofIvy]

If you are allowed to use "the sum of two rational numbers is rational" then the problem is easy: If $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k$ is rational, then $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k+ r_{k+1}$$= (r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k)+ r_{k+1}$, the sum of two rational numbers.

Of course, you can prove that the sum of two rational numbers is rational by:
$$\frac{a}{b}+ \frac{c}{d}= \frac{ad+ bc}{bd}$$.

 

[willbland]

If you're ever in a situation where n is not equal to n, induction will be the least of your worries.

It doesn't sound like you completely understand what induction is supposed to be saying. You have to do two things: Prove the statement is true when n=2, which it sounds like you did, and prove that if the statement is true when n=k, it's true when n=k+1

For example, can you prove that the sum of three rational numbers is rational, using the fact that the sum of two rational numbers is rational? Can you prove that the sum of four rational numbers is rational, using the fact that the sum of three rational numbers is rational? Try to generalize this this for any k

I'm kind of confused by this post.

Here's my understanding of induction,
Your proving that it's true for when n = 2, then your assuming it's true for any number n, then you have to prove that if n is true, then n + 1 is true, and so on.

Your basically proving that it's true for all n > 1 because of the domino effect. I'm still troubled by how to prove n + 1 in this particular proof.

If you are allowed to use "the sum of two rational numbers is rational" then the problem is easy: If $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k$ is rational, then $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k+ r_{k+1}$$= (r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k)+ r_{k+1}$, the sum of two rational numbers.

Of course, you can prove that the sum of two rational numbers is rational by:
$$\frac{a}{b}+ \frac{c}{d}= \frac{ad+ bc}{bd}$$.

I did prove that the sum of two rational numbers was rational that exact way, but i don't know how to take it to the next step and prove that n + 1 is rational :(

So far I've just proven that when n = 2, the sum is rational,

now i assume when n = k the sum is rational, and have to prove that when n = k + 1 the sum is rational, but I'm completely lost when it comes to doing this for this particular proof. :(

 

[Mark44]

I'm kind of confused by this post.

Here's my understanding of induction,
Your proving that it's true for when n = 2, then your assuming it's true for any number n, then you have to prove that if n is true, then n + 1 is true, and so on.

Your basically proving that it's true for all n > 1 because of the domino effect. I'm still troubled by how to prove n + 1 in this particular proof.

Part of your confusion might be in the terminology. There's a statement in this and every induction proof, that depends on n in some way. For this problem, it is "The sum of n rational numbers is a rational number."

It doesn't make a whole lot of sense to talk about the sum of one rational number, so we can assume that n is at least 2. For each value of n >= 2, we get a different statement. E.g,
S(2): The sum of 2 rational numbers is a rational number.
S(3): The sum of 3 rational numbers is a rationa number.
.
.
.
S(k): The sum of k rational numbers is a rational number.
S(k + 1): The sum of k + 1 rational numbers is a rational number.
.
.
.

You're not trying to "prove n + 1"; you're trying to prove that the n + 1st statement is true, given that the nth statement is true.

I did prove that the sum of two rational numbers was rational that exact way, but i don't know how to take it to the next step and prove that n + 1 is rational :(

So far I've just proven that when n = 2, the sum is rational,

now i assume when n = k the sum is rational, and have to prove that when n = k + 1 the sum is rational

When n = k, you are assuming that r₁ + r₂ + ... + r_k is a rational number.

Now, what statement are you trying to prove when n = k + 1, and how can you use your assumption to do this?

l, but I'm completely lost when it comes to doing this for this particular proof.

:(

 

[HallsofIvy]

Notice that I said, before, :"if $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k$ is rational, then $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k+ r_{k+1}$$= (r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k)+ r_{k+1}$ is the sum of two rational numbers".

I intentionally used "k" rather than "n" to avoid confusing the "induction step" with the statement in the theorem. But the important point is that since $r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k$ is a rational number, then $= (r_1+ r_2+ r_3+ \cdot\cdot\cdot+ r_k)+ r_{k+1}$ is just the sum of two rational numbers, and so rational!