Maths Puzzle 16 - Arithmetic Progression

A friend gave me this sum to solve.

One day John went to the woods and he wanted to pick up some cones. Under the first tree he found 11 cones, under the second one was 6 more cones than under the first one, under the third one he found 6 more cones than under the second one and so on... How many cones did John find if in all together if he went under 77 trees?

Now this is a classic Arithmetic Progression (AP) problem.

An AP series is like:

a, a+r, a+2r, a+3r .... (a+(n-1)r..

where a is the 1st term, there are n terms and the difference between the each term and the next term is constant = r.

The sum of such an AP series can be found as follows:

Total cones T = a   +                 (a+r)     +   (a+2r) +  ......       + (a+(n-1)r)                     

                T = (a+(n-1)r   + (a+(n-2)r) +  (a+(n-3)r) ....            +     a                         (same as above in reverse order)

                  -------------------------------------------------------------------------------

Adding both 2T= (2a+(n-1)r) +(2a+(n-1)r) +  (2a+(n-1)r) +  .... + (2a+(n-1)r)  

        2T = (2a+(n-1)r)*n

T = (2a+(n-1)r)*n/2

now a = 1st term = 11

r = difference = 6

n = 77

T = (2*11 +(77-1)*6)*77/2

   = (22+76*6)*77/2

   = 478*77/2

   = 18403