Question

PLEASE! IM BEGGING!! HELP ME!! PLEASE!! *TEARS*
Can someone please help me with this? I post it earlier but the answer wasnt what my teacher wanted. Can someone help me I will post in attachment.

Answer 1

we want to prove that the formula
3 + 5 + 7 + ... + 2n+1 = n(n+1) is true for positive integers n=1,2,3,...
the problem is, there are an infinite number of statements to check, and we will never get done, one for each number n.
so we use mathematical induction

Answer 2

Im following

Answer 3

I am going to rewrite this as a proposition P(n)
Let P(n) be the statement 3 + 5 + 7 + ... + 2n+1 = n(n+1)

Answer 4

okay

Answer 5

mathematical induction says :
If you have a statement that is true for some number, for example n=1
(this is called the basis case)
and furthermore that it can be proven if P(k) is true, then P(k+1) is true.
then it must be the case that P(n) is true for all n greater than your basis

Answer 6

this is the principle of mathematical induction.
I didnt actually do your example yet, this just preliminary

Answer 7

I know, thank you, good notes

Answer 8

so for instance if your basis case P(1), and assuming you showed P(k)->P(k+1)
then you have P(2) because P(1) is true and P(1) ->P(2) is true
then it must be true P(2) is true , etc

Answer 9

ok thats just the logic behind it.
so in our example P(n) : 3 + 5 + 7 + ... + 2n+1 = n(n+1)
is P(1) true?

Answer 10

notice on the left side, you are summing a sequence of odd numbers 3,5,7,...The formula for the sequence of odd numbers is 2n+1 ,

Answer 11

so we can rewrite the statement P(n)

Answer 12

ok im starting to understand