Progressions

Most of the points that I have listed below are very basic.

In a Arithmetic Progression a,b,c.
The arithmetic mean b = (a+c)/2.

If we insert 'n' arithmetic means between 2 terms a and b,
we will get an AP with n+2 terms.

Take 2 numbers 10 and 20.
If we are inserting 1 Arithmetic mean between these 2 terms,
it will be 15.
Now 15 is the Arithmetic mean of 10 and 20.
If we are inserting 3 Arithmetic means between 10 and 20.
The AP becomes
10,12.5,15,17.5,20
Now the Arithmetic mean of the 3 inserted numbers 12.5,15,17.5 is
15 which is the arithmetic mean of the original 2 numbers.
Here in the AP 10,12.5,15,17.5,20 we have 5 terms,

To determine the arithmetic means to be inserted,
we can use the formula
(Difference between the 2 terms)/Number of means to be inserted+1

Lets see the case explained above.
We need to insert 1 AM between 10 and 20.
So 20-10/(1+1)
We get 5.
We need add this to the smaller number.
We get 15.

We see that in an AP with odd number of terms,
first term+last term = second term + second last term = 2*middle term

Some properties that will of help while solving problems.
If Sum of 'P' terms = Sum of 'Q' terms then
Sum of 'P+Q' terms = 0

Consider the AP 2,1,0,-1,-2
Here Sum upto 1 term is 2
Sum of 4 terms is also 2.
Sum of the 5 terms is 0.

If 'P'th term=Q and 'Q'th term=P
'P+Q'th term = 0

Consider 4,3,2,1,0
4th term is 1
1st term is 5
and 5th term is 0