Use mathematical Induction to prove the formula for all integers n with the given values:
11+15+19...+(4n+7)=2n^2+9n , n≥1
Then you assume it's true for , for some , and from that, you try and derive the truth of that expression for .
So by assuming that is true (this is your induction hypothesis), you want to prove that:
The way you would do that is to say that:
using the induction hypothesis.
Now you want to show that the RHS is .
Match up what I've said above with what your text book says about proof by induction and try and work out the why of it rather than the what.
It works like this.
You've got a proposition involving n that you're trying to trying to prove is true for all n. (The usual example is ).
First you show it's true for n=1 (or n=0, or whatever, depends on the expression). In the above case, we have . This is called the "basis for the induction (or the "base case").
Then you say: "Suppose this proposition is true for , where is any number . Let's just, for a time being, assume that it's true." In the above example, we see that means . This is called the "induction hypothesis".
Now, if by assuming that it's true for , we can prove it's true for , we know that it will then be true for all . This is called the "induction step".
In the above, we want to show that .
And we do that by saying:
(we've used the induction hypothesis here, which we temporarily assumed true above)
(by some fairly straightforward algebra)
And so we have shown that:
IF is true,
THEN is true.
And we already know that it's true when k=1.
So, "by the principle of mathematical induction", or "by mathematical induction", or just "by induction", is true for ALL values of n.
See how it works? Now see how I've applied the above to the question you asked.
It's true for 1. And, if it's true for , then it's true for .
Because it's true for 1, it's true for 1+1=2. And because it's true for 2, it's true for 2+1 = 3. And because it's true for 3, it's true for 3+1 = 4. And so on. Thus it's true for all numbers, by the fact that the numbers all go 1, 2, 3, 4, ... and so on "to infinity".
There are several underlying philosophical questions underlying the validity of the above. So much so, that the "principle of induction" is taken as an axiom (that is, a "postulate", or "basic truth") of the system of natural numbers. So don't take it too much to heart if you're not sure it "makes sense" or feel that the reasoning seems a little flimsy - the logical justification is a deep question.
so from your example n=1 so (4)(1)+7=2(1)^2+9(1) which = 11=11 so true
i assume that n=k , 11+15+19+...+(4K+7)=2K^2+9k is true
now i must prove for (k+1) 11 +15 +19+...+(4(k+1)+7)=2(k+1)^2+9(k+1)
now i add the (4k+7) to both sides from the induction hypothesis
no this is where in stuck now.. what is next and how do i finish it. lol its like 3:18 am where in at. im dead tired.