How would you represent a continuous sum of numbers, such as 13+16+19+22+25 using summation notation?

To find out how the numbers are related, put them as a sequence :

13, 16, 19, 22, 25

Find the first differences :

16 - 13 = 3

19 - 16 = 3

22 - 19 = 3

25 - 22 = 3

Here, all differences are the same.

So they can be represented by a linear equation, say, y = ax + b.

(If the second differences were the same, you would need a quadratic equation.)

13 is the 1st term, so if x = 1, y = 13. The equation becomes : 13 =a + b.

This has two unknown variables, so we need a second equation to solve for both.

16 is the 2nd term, so if x = 2, y = 16. This equation is : 16 = 2a + b.

We have :

(1) a + b = 13

(2) 2a + b = 16

Solving these simultaneous equations gives : a = 3 and b = 10.

So each term belongs to the formula, y = ax + b = 3x + 10.

The sum of the series, 13 + 16 + 19 + 22 + 25 is then :

..... 5

..... ∑ (3x + 10)

... x=1