Hi
The first step consists in verifying that the statement is true for n=1
For the inductive step you assume that the statement holds for the case n = k
And you need to show that
Hint :
I just started reading about number theory and here was the first problem verbatim.
Remarks and Hints: A statement which is proved by induction often has an integral parameter,such as n in our case. You then need to do two steps, either of which can be done first:(1) You prove that the statement holds for the case n = 1, or whatever the first case isSo how do I prove this exactly? I know that if I sum j from 0 -> n where n=1 I'll get one, and i know for n=1 that n(n+1)/2 = 1(1+1)/2 = 1 so that makes it true, but what about for the inductive step? What do I do for n=k or n = k+1 ?
that you’re interested in. This is called the base case.
(2) You prove that IF the statement holds for the case n = k, THEN it holds for the case
n = k + 1. This is called the inductive step.
Thus, for step 2, you get to ASSUME the statement for n = k, and show that this is enough
to imply it for n = k + 1. In your proof, indicate clearly when you’re doing the base case
and when you’re doing the inductive step.
(1) Suppose that you have shown the statement for n=1
(2) Suppose that you have shown that IF the statement holds for a certain value k THEN it holds for (k+1)
Due to the fact that it is true for n=1 (statement (1)), then it is true for n=2 (statement (2))
Due to the fact that it is true for n=2, then it is true for n=3 (statement (2))
etc
This shows that the statement is true for every n
This is the methodology
Now let's prove the inductive step. IF the statement holds for the case n = k
Then
Therefore the statement is true for (k+1)
Here you said the following:
"Due to the fact that it is true for n=1 (statement (1)), then it is true for n=2 (statement (2))Due to the fact that it is true for n=2, then it is true for n=3 (statement (2))"But in your post before that I saw an example of n=1 and n=k, then n=k+1. I do not see what you are saying for n=2 and n=3 and how you are referencing things to statements(#). (Sorry, I am green to this course, trying to learn)
Secondly, where did this come from or how did you make this happen:
I'm not up 100% on my summer operator. I'd like to understand how that worked above
First you show that the statement is true for n=1.
Then you show that if it is true for a certain integer k then it is true for (k+1). The first part of this sentence is true for k=1, therefore it is true for 1+1=2 OK?
Now that it is true for k=2, it is true for 2+1=3
etc
Therefore
Suppose that the statement is true for k
Then
Therefore
Okay, I guess I need to clarify my questions a little more:
I understand these statements:
But I do not understand this statement:
Therefore
How does this happen? Do you mean this is the same as:
And lastly, how did this get formed exactly:
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I'm curious as to if there is distribution or something with the Sum operator.