Finding the nth term in a "backwards" sequence isn't about reversing the typical order, but rather understanding that we can still apply the same fundamental principles and formulas even when a sequence appears to be decreasing or 'backward'. The key is correctly identifying the type of sequence and applying the appropriate formula with attention to how the values are changing.
Understanding Sequence Types
Before we calculate the nth term, we must identify whether the sequence is arithmetic or geometric.
Arithmetic Sequences
An arithmetic sequence has a constant difference between each term. The formula for finding the nth term (an) in an arithmetic sequence is:
an = a + (n − 1)d
Where:
- an is the nth term
- a is the first term
- n is the term number
- d is the common difference
Note: The common difference, d, can be negative if the sequence is decreasing.
Geometric Sequences
A geometric sequence has a constant ratio between each term. The formula for finding the nth term (an) in a geometric sequence is:
an = arn−1
Where:
- an is the nth term
- a is the first term
- r is the common ratio
- n is the term number
Note: The common ratio, r, can be less than 1 if the sequence is decreasing.
Applying the Formulas to "Backwards" Sequences
The term "backwards" often refers to sequences that appear to be decreasing, but the core method to find the nth term is still the same: determine the pattern and apply the corresponding formula with the correct values.
Example 1: Arithmetic "Backwards" Sequence
Let's look at the arithmetic sequence: 10, 8, 6, 4...
- Identify the type: This is an arithmetic sequence because the difference between each term is constant (8-10=-2, 6-8=-2, etc.)
- Find the common difference (d): Here, d = -2
- Identify the first term (a): a = 10
- Apply the formula: an = 10 + (n-1)(-2)
- Simplifies to an = 10 - 2n + 2
- Final Formula: an = 12 - 2n
Therefore, to find the 5th term (a5):
a5 = 12 - 2(5) = 12-10 = 2
The 5th term would be 2, continuing the sequence 10, 8, 6, 4, 2,...
Example 2: Geometric "Backwards" Sequence
Consider the geometric sequence: 16, 8, 4, 2...
- Identify the type: This is a geometric sequence because the ratio between each term is constant (8/16 = 1/2, 4/8 = 1/2, etc.)
- Find the common ratio (r): Here, r = 1/2
- Identify the first term (a): a = 16
- Apply the formula: an = 16 * (1/2)^(n-1)
Therefore, to find the 5th term (a5):
a5 = 16 (1/2)^(5-1) = 16 (1/2)^4 = 16 * 1/16 = 1
The 5th term would be 1, continuing the sequence 16, 8, 4, 2, 1...
Key Takeaway
The important thing to remember is the formulas for finding the nth term do not change based on whether a sequence is increasing or decreasing. The only thing that changes are the values for the common difference d (for arithmetic) or the common ratio r (for geometric), which will be negative or a fraction less than one in decreasing sequences.
