this is the equation, I don't understand how they got from a to b
Hi, I understand the sequence of the example I am working through, but I cannot understand some of the "simplification" (i.e. when one expression is simplified to mean the same thing)
The example is
Prove inductively for all integers n >= 1,
1+2+3....+n = ---> *
i) i understand s(subscript m ??) is true, I prove this by substituting an integer for n
ii) sn+1 is true when n>=m and sn are true
To prove for Sk+1 is true for n = k+1;
original k(k+1)/2 becomes
k(k+1)/2 + (k+1) ---I understand this is proof for n=k+1 where n is the formula above, marked ---> *
but I do not understand the next step in the example, where
(a) k(k+1)/2 + (k+1) <---this formula is simplified to:
= (b) (k+1)(k+2) / 2 <--- I don't understand how they went from (a) to (b) here.
k(k+1)/2 + (k+1) = k squared + k / 2 + (k+1) (sorry I can't get super/sub script to work)
Thanks for any help, sorry if my explanation seems jumbled or unreadable(if so I will try to post the example again more clearly)
I didn't think I had forgotten that much algebra, but ...
oh I get it now, working backwards I see that
I just didnt associate (k+1)(k+2) with
off the top of my head.
I will go back over my notes and hope I "get" the problem better next time through.
Thanks for the help.
BTW should I have posted this in the algebra formula?
It's common in these induction problems because we try to establish an
infinite chain of cause and effect from the first term all the way through to infinity.
We try to discover that the formula being valid for n=1 causes it to be valid for n=2...
then valid for n=2 causes it to be valid for n=3...
valid for n=3 causes it to be valid for n=4...
valid for n=4 causes it to be valid for n=5....
all the way to infinity.
If we can prove that being valid for n=k causes it to be valid for n=k+1,
then we have proven all the foregoing in one step!!
Since this proves that any term k causes the formula to be valid for the next term k+1
So the induction step establishes whether or not this infinite chain exists.
In this case
substitute k+1 for k
We try to prove that P(k) being true causes P(k+1) to be true
Hence, we can either multiply out
and check that it's the same as multiplied out
or factor to see if it equals