Explainer: Geometric Series

In Core 1 you learn about arithmetic sequences, where the pattern is based around an addition/subtraction rule. For example, a sequence with a common difference of 2
(5, 7, 9, 11, 13) is an arithmetic sequence. This chapter focuses on sequences where the pattern is based around multiplication or division.
A sequence such as 2, 4, 8, 16, 32 is known as a geometric sequence.
You need to be able to spot patterns to work out the rule for a geometric sequence. In any geometric sequence, u₁, u₂, u₃,..., u_n

u_n+1

---

u_n

is the common ratio, and is the same for all consecutive pairs

Example 1
Calculate the common ratio for each of the following sequences :
a) 3, 12, 48, 192
Answer

b) 54, 18, 6, 2
Answer

You can define a Geometric Sequence using a first term "a" and a common ratio "r"

Example 2
Find the nth and 10th terms of the following sequences : 3, 6, 12, 24,

Application of Geometric sequence- investing in the bank
Gerry invests 100 pounds at 5% yearly interest rate. How many years does he need to wait until his investment doubles.

	Find a and r
	Use logs to find n
	Solve to find n

Notice that if the interest rate is 0, Gerry's initial investment will still only be worth 100 pounds. You may also wish to see how Gerry's investments grows at an interest rate of 5%.

You need to be able to work out the sum of a Geometric Sequence
Click on each stage to see how the formula for the sum of the geometric series is derived.

	Multiply by r
	Subtract first equation from the second equation
	Cancel terms
	Rearrange to find Sn
	Factorise out the 'a' and 'r'

Refresh

Example 3
Find the sum of the following series: 1+2+4+8+....+1024

	Find r
	Find a
	Find n
	Use logs to find n

Refresh

Example 4
Find the sum of the following series: 1024 + 512 + 256 + 128 + ....... + 1

	Find r
	Find a
	Find n
	Use logs to find n

Notice that the first and second series are the same. The first one can be seen as an increasing geometric series where r > 1 and the second series is the decreasing gemetric series with -1<r<1

Application of Geometric series- investing in the bank
Gerry invests 100 pounds at the beginning of every year, at 5% yearly interest rate for 10 years, find the value of his capital after 10 years.

	Find a, r and n
	Use geometric series formula
	Compute

Alternativaly you may decide that a = 100(1.05)¹⁰ and r = 1.05 ^-1 . Experiment for yourself, you will see that you arrive at the same answer. You may also wish to experiment with different interest rates. Notice that if the interest rate is 0 then the future value would be 10 times 100 pounds = 1000 pounds. As the interest rate increases the future value will also increase.