Geometric Sequences
Neil Trivedi
Teacher
Geometric Sequences
A geometric sequence is one that increases/decreases by a common ratio. This means to go from term to term you are multiplying.
Notation:
- The first term
- The common ratio
- The term number
Formula:
For a geometric sequence, each term is defined deductively as,
A geometric sequence with the terms can be written as .
Example 1:
The second and fifth terms of a geometric sequence are and , respectively. Find the common ratio and the first term of the series.
Step 1: Write the third and fourth term in terms of and .
Step 2: Solve the simultaneous equations by dividing the two equations to eliminate .
Substitute into either or to find . We will use .
No answer provided.
Example 2:
What is the first term in the geometric progression to exceed million?
Step 1: Recognise the first term , common ratio , and the inequality needed.
Step 2: Solve the inequality.
Divide both sides by .
Take logs of both sides.
Bring down using our power rule for logarithms.
Divide both sides by .
No answer provided.
Geometric Series
Geometric Summation Formula:
For a geometric sequence, the summation of terms is defined deductively as,
where is the first term, is the common ratio and is the number of terms.
Proof (Must be known):
We write and out, and find the difference:
All terms will cancel now, apart from on the first line, and on the second. We will be left with:
Example 3:
The first three terms of a geometric sequence are , , and respectively, where is a positive constant.
a) Find the value of .
Single Step: We know a geometric sequence has common ratio between terms, so we equate the ratio between terms.
Solve using the quadratic formula.
or (rejected as is given to be positive in the question)
Therefore, .
b) Find the sum of the first terms of the sequence.
Step 1: Find and .
Step 2: Apply the geometric summation formula.
No answer provided.
Summing to Infinity
Summing to Infinity Formula:
Condition:
It is essential to have , so that we have a convergent sequence a sequence that goes to a certain value as tends to infinity, so that we can sum it to infinity. If the sequence is divergent , the value of each term keeps increasing and the sum will just be infinity.
Example 4:
The first term of a geometric sequence is and the sum to infinity of the series is .
a) Find the common ratio of the sequence.
Single Step: Substitute the values to the summing to infinity formula and solve it.
b) Find the fourth term of the sequence.
Single Step: Apply the geometric sequence formula.
c) Find the exact difference between the sum of the first four terms of the series and the sum to infinity.
Step 1: Find the sum of the first five term .
Step 2: Find the difference.
No answer provided.
Example 5:
The first term of a geometric sequence is and the common ratio of the series is .
a) Find the first term to have value less than .
Step 1: Recognise the first term , common ratio and the inequality needed.
Step 2: Solve the inequality.
Since is negative, we flip the inequality when we divide both sides by .
b) Find the sum to infinity of the series.
Single Step: Apply the sum to infinity formula.
No answer provided.
Challenging Question