Question

Write the sum using summation notation, assuming the suggested pattern continues. -9 - 4 + 1 + 6 + ... + 66

Answer 1

Summation notation is:

`\sum_(n=1)^(16)[5(x-1)-9]`

If you prefer it a little more simplified:

`\sum_(n=1)^(16)(5x-14)`

Explanation:

First my favorite part, finding a pattern between the consecutive terms.

This is an arithmetic series because the terms are going up by 5 each time.

So arithmetic sequence, think linear equations:

x | y

1 -9

2 -4

3 1

4 6

..................

n 66

We are going to have to find that n but will will eventually...

The equation for a line in point slope form is
`y-y_1=m(x_x_1)` where
`(x_1,y_1)` is a point on the line and m is the slope.

We are already have the slope is 5 (the slope is the common difference in arithmetic sequence).

I'm going to use the first point (1,-9).

So the equation in point slope form is
`y-(-9)=5(x-1)`

Subtract 9 on both sides:

`y=5(x-1)-9`

Now we need to know how many terms we are adding so what is x if y=66.

`66=5(x-1)-9`

Add 9 on both sides:

`75=5(x-1)`

Divide both sides by 5:

`15=x-1`

Add 1 on both sides:

`16=x`

We have 16 terms in this sequence where the 16th term is 66.

Summation notation is:

`\sum_(n=1)^(16)[5(x-1)-9]`

You could simplify the 5(x-1)-9.

Distribute: 5x-5-9

Add like terms: 5x-14

Answer 2

Explanation: