Question

5. Create a rule (or equation) for the following geometric sequence 25, -100, 400, -1600 . . .

Answer 1

Geometric Sequence

In a geometric sequence, each term is calculated as the previous term times a constant number called the common ratio.

If we are given the first terms of the sequence, we can find the common ratio by dividing two consecutive terms.

We are given the sequence:

25, -100, 400, -1600

Dividing the second by the first term:

`r=(-100)/(25)=-4`

Dividing the third term by the fourth term:

`r=(400)/(-100)=-4`

The common ratio is proven to be constant.

The formula to calculate the nth term of a geometric sequence is:

`a_n=a_1\cdot r^(n-1)`

Where a1 is the first term. In our problem a1 = 25.

The rule or equation for the given geometric sequence is:

`a_n=25\cdot(-4)^(n-1)`