How do I find the sum of the following numbers?

Question

How do I find the sum of the following numbers:

3+7+11+15+19+. . . . +403

Answer 1

Long winded method: work out all the numbers [each number adds 4] then add them altogether. There probably is an easier method but I can't think of it.

Answer 2

First, you have to identify the pattern in these numbers.  The pattern:

3 = 4*(0)+3

7 = 4*(1)+3

11 = 4*(2)+3

15 = 4*(3)+3

...

403 = 4*(100)+3

In this case, you are summing up (4*n+3), where n = 0,1,...100.  I believe that you are learning the summation notation.  The mathematical equation is:  

summation of (4*n+3) from 0 to 100

= 3 + summation of (4*n+3) from 1 to 100

= 3 + summation of (4*n) from 1 to 100 + 3 *(100)  <-- adding 3 for 100 times

= 3+ 4 * (summation of n from 1 to 100) + 3*(100)  <-- let's say if you have 4a+4b, you can write it as 4(a+b).

there is a theorem that says, summation of n from 1 to k = (k*(k+1))/2, so your equation will become

= 3+ 4 * ( ((100)*(100+1)/2) ) + 3*(100)

= 20503

It's hard to explain by typing in a computer, but I hope it helps you understand!  

Answer 3

add them up...

3+7=10

10+11=21

21+15=36

36+19=55

55+403=458