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

Summation notation is:

If you prefer it a little more simplified:

Step-by-step 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 where 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

Subtract 9 on both sides:

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

Add 9 on both sides:

Divide both sides by 5:

Add 1 on both sides:

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

Summation notation is:

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

Distribute:  5x-5-9

Add like terms:  5x-14

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.