1. Geometric Sequences and Series
Summation

2. The sequence
is an example of a
Geometric sequence
A sequence is geometric if
where r is a constant called the common ratio
In the above sequence, r = 2

3. A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is

4. Exercises
1. Use the formula for the nth term to find the term indicated of the following geometric sequences
(a)
Ans:
(b)
Ans:
(c)
Ans:

5. The formula will be proved next but you don’t need to learn the proof.
Summing terms of a G.P.
e.g.1 Evaluate
Writing out the terms helps us to recognize the G.P.
With a calculator we can see that the sum is 186.
But we need a formula that can be used for any G.P.
The formula will be proved next but you don’t need to learn the proof.

6. Summing terms of a G.P.
With 5 terms of the general G.P., we have
TRICK Multiply by r:
Subtracting the expressions gives
Move the lower row 1 place to the right

7. Summing terms of a G.P.
With 5 terms of the general G.P., we have
Multiply by r:
Subtracting the expressions gives
and subtract

8. Summing terms of a G.P.
With 5 terms of the general G.P., we have
Multiply by r:
Subtracting the expressions gives

9. Summing terms of a G.P.
So,
Take out the common factors
and divide by ( 1 – r )
Similarly, for n terms we get

10. Summing terms of a G.P.
gives a negative denominator if r > 1
The formula
Instead, we can use

11. Summing terms of a G.P.
For our series
Using

12. Summing terms of a G.P. EX
Find the sum of the first 20 terms of the geometric series, leaving your answer in index form
Solution:
We’ll simplify this answer without using a calculator

13. Summing terms of a G.P.
There are 20 minus signs here and 1 more outside the bracket!

14. 3rd term + 4th term = 4( 1st term + 2nd term )
Summing terms of a G.P.
e.g. 3
In a geometric sequence, the sum of the 3rd and 4th terms is 4 times the sum of the 1st and 2nd terms. Given that the common ratio is not –1, find its possible values.
Solution: As there are so few terms, we don’t need the formula for a sum
3rd term + 4th term = 4( 1st term + 2nd term )
Divide by a since the 1st term, a, cannot be zero:

15. Summing terms of a G.P.
We need to solve the cubic equation
Should use the factor theorem:
We will do this soon !!

16. Summing terms of a G.P.
The solution to this cubic equation is therefore
Since we were told we get

17. SUMMARY
A geometric sequence or geometric progression (G.P.) is of the form
The nth term of an G.P. is
The sum of n terms is or

18. Sum to Infinity
IF |r|<1 then
Because (<1)∞ = 0

19. 2. Find the sum of the first 15 terms of the G.P.
Exercises
1. Find the sum of the first 15 terms of the following G.P., giving the answers in index form
2. Find the sum of the first 15 terms of the G.P.
giving your answer correct to 3 significant figures.

20. 2 + 8 + 32 + . . . 4 - 2 + 1 + . . . Exercises 1. Solution:
2. Solution:
( 3 s.f. )