1. Sequences and Series
Geometric sequence.

2. What is a Geometric Sequence?
In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio r .

3. For example, look at this sequence:
Some sequences increase or decrease by multiplying or dividing each term by a constant factor, which we call a common ratio r.
For example, look at this sequence:
2, , , , , , , 256, . . . ×2 ×2 ×2 ×2 ×2 ×2 ×2
This sequence starts with 2 and increases by multiplying the previous term by 2.
Stress that this sequence increases in unequal steps. It is not linear.
If we looked at the differences between the consecutive terms they would be 2, 6, 8, 16, 32 … in other words, they would form the same sequence.
This is a sequence of powers of 2. We could write it as 21, 22, 23, 24, 25, 26, 27 …

4. Can you work out the next three terms in this sequence?
1024, 256, 64, 16, , 1,
0.25,
0.0625, ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 ÷4
How did you work these out?
This sequence starts with 512 and decreases by dividing by 4 each time.
Each term in this sequence is one quarter of the term before.
We could also continue this sequence by multiplying by 1/4
each time.

5. Example 1: Determine if the sequence is geometric
Example 1: Determine if the sequence is geometric. If so, identify the common ratio
1, -6, 36, -216
Yes. r = -6
2, 4, 6, 8
No. No common ratio.

6. Geometric Sequence
A geometric sequence is one where to get from one term to the next you multiply by the same number each time. This number is called the common ratio, r.
2, 10, 50,
r = 5 x5 x5 x5

7. Important Formula for geometric sequence:
Formula for the nth term
where:
un is the nth term in the sequence
u1 is the first term
n is the number of the term
r is the common ratio

8. Example 2. Find the 19th term in the sequence of 11, 33, 99, 297 . . .
un = u1 r n-1
Start the formula
Find the common ratio between the values.
Common ratio = 3
u19 = 11 (3) (19-1)
Substitute known values
u19 = 11(3)18 =4,261,625,379

9. Example 3. How many terms are there in the geometric sequence 2, 6, 18, …., 1458?
nSolve on GDC
7 terms

10. Example 4. Find the first term of a geometric sequence 1, 5, 25, 75, …
Example 4. Find the first term of a geometric sequence 1, 5, 25, 75, ….. which exceeds
The 8th term.
Check.

11. Example 5. The second term of a geometric sequence is
-15 and the fifth term is 405. Find the first term and the common ratio.
Use GDC to solve.
Divide eq(2) by eq(1) side by side:

12. Example 6. In a geometric sequence, the third term is 3 and the seventh term is 48. Find the 10th term.
Divide eq(2) by eq(1) side by side:
Use GDC to solve.
Numerical solve will only give one answer, so graphical solve is the way to go.

13. when r = 2
when r = - 2  

14. 300 is at the start of the first year, it is not the first term
300 is at the start of the first year, it is not the first term. The first term is after one year.
Check:

17. Example