Sorry this is wrong. O.o
Hey guys, im really struggling with induction. Question: Use induction to prove that 1 + 7 + 19 +..... + ( ) = for all positive integers by completing the following steps:
a) Show that the statement is true when .
My answer: Is this just substituting into ? and how do I know its true?
b) show that, if 1 + 7 + 19 + .... + ( ) = , then 1 + 3 + 5 + ... + ( ) + ) = .
my answer: Not sure this one! Any help would be much appreciated.
Part b) attempts to show that if the statement is true for some n=k,
then being true for n=k causes the statement to be true for the next n,
which is n=k+1.
The purpose of this is to show that...
true for n=1 causes the equality to be true for n=2,
true for n=2 causes the equality to be true for n=3,
true for n=3 causes the equality to be true for n=4....
all the way to infinity
then you can see by thinking about it that an infinite chain of cause and effect
has been set up between adjacent terms.
Hence we express the k+1 version in terms of the k version
to see if there is a cause and effect relationship.
Essentially, this is why part b) is written as it is.
?
If it is then examining
which is the sum of (k+1) terms,
and examining if this is if the sum of k terms is we get
which is since
Hence, the equality being true for some term n=k causes the equality to be true for the next term n=k+1.
Therefore there is an infinite chain of cause and effect.
If the equality is true for n=1, this causes the equality to be true for all n.
The reason is...
if
then the sum of the first k terms is
This means that the sum of the first terms will be
if this equality really is true!
Think about that.
However, the sum of the first terms is the sum of the first k terms plus the (k+1)th term, which is
if the sum of the first k terms really is
You get the (k+1)th term by using k+1 in place of k.
Then you need to show that when you add the sum of k terms (which is supposed to be ) to the (k+1)th term,
the answer should be
If this is the case, then we can say
causes the to be
All we've done is to show that the formula being true for a particular value of n=k
will cause the formula to be true for the next value of n=k+1.
That establishes a chain reaction.
Then if you test it for n=1, and it's true, then it's true for all natural numbers n.
Since this may seem long-winded, it's generally not covered in class.
There are shortcut ways to prove the initial statement,
such as
test for n=1
assume p(k) true.....test p(k+1)
but it's easier if you understand the process.