This is an arithmetic series because the difference between
any term (after the first) and the term that precedes it is
4.  So the common difference is given by .

The formulas for an arithmetic series are

1.  For the nth term:



where  is the first term

2.  For the sum of the terms through the nth term:



In your problem, ,  and , so

we use the first formula to find n:








So .

Then we substitute in the second formula:









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

Here is another way to do it without using formulas:

S = 4 + 8 + 12 + 16 + 20 + ...... + 100 

S = 4(1 + 2 + 3 + 4 + 5 + ...... + 25) 

so there are 25 terms:

Write the sum, S, forward, then backwards

S =   4 +  8 + 12 + 16 + 20 + ...... + 100
S = 100 + 96 + 92 + 88 + 84 + ...... +   4

Add the two equations term by terms:

 S =   4 +   8 +  12 +  16 +  20 + ...... + 100
 S = 100 +  96 +  92 +  88 +  84 + ...... +   4
-----------------------------------------------
2S = 104 + 104 + 104 + 104 + 104 + ...... + 104

Since there are 25 terms, there are 25 104's above, so

2S = 25*104
2S = 2600
 S = 1300

Your teacher probably wants you to do it the first way, though.

Edwin