How to Find the Formula of Arithmetic and Geometric Sequences
The arrangement of numbers in a particular order is known as a sequence. Sometimes, we are asked to find the value of a specific term in a sequence. One method to find the value of a specific term in a sequence is to extend the sequence until we reach the desired term, Next approach is to determine the sequence finding rule of the nth term of a sequence and then calculate the term you need.
Extending a sequence to find the missing term in a sequence is not always a realistic approach. For example, we cannot extend a sequence from the beginning to find a value 300th term. In this case, determining a sequence finding rule provides a more elegant and efficient way to find unknown terms in a sequence. Sequence rule finder is an online tool that enables you to find the unknown term in a sequence efficiently.
Term to Term Rule
A term to term rule enables you to find the next term in a given sequence if you know the previous term This is also known as a recursive rule. For example, if the given sequence is 2, 4, 6, 8, then to find the next term you can use the general formula: an = an-1 + 2 (a5 = a4 + 2). The drawback of the term-to-term rule is that you should know the previous term to calculate the next term.
General Rule of Arithmetic Sequence
Given a sequence with the first term a₁ and the common difference d, the nth or general rule of an arithmetic sequence is given by an = a1 + (n - 1)d.
Example:
Find the 10th term of an arithmetic sequence 5, 8,11, 13
a1 = 5 , d = (8 - 5) = 3
Accordingly.
a10 = 5 + (10 - 1)3
a10 = 5 + (9)3
a10 = 32
Explicit Rule
The explicit rule, also known as the position-to-term rule, allows you to calculate the value of any term. For example, in 2, 4, 6, 8…. the first term is 2, the second term is 4, the third term is 6, the fourth term is 8, and to calculate the fifth term here we use the formula an = 2n = 2(5) = 10. Hence, the fifth term here is 10. The 100th term is 2(100) = 200.
Solved Example
1. Write a rule for the nth term of the sequence given below.
1, 4, 9, 16, 25, 36,?
Solution:
Each term in the sequence follows the pattern 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25…
These are the squares.
To find the nth term of a given sequence, we will follow the following rule.
Rule = an = n²
Here, an is “term number n”.
Accordingly, the 7th term of the given sequence is a7 = 7² = 49.
2. Find the next term of the sequence: 1, 2, 5, 10, 17,?
Solution:
In the given sequence, we can see each term is increasing. Taking the difference between adjacent terms we get:
2 - 1 = 1
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
Here, we can see the difference between the adjacent terms is odd numbers. Accordingly, the next term must be 9.
Hence, the next term in the sequence is 17 + 9 = 26.
FAQs on Finding the Rule of a Sequence Step by Step Guide
1. What is the rule of a sequence in maths?
The rule of a sequence is a formula or pattern that explains how to find each term in the sequence. It tells you how the numbers are generated and how to calculate any term.
- A sequence is an ordered list of numbers.
- The rule can be written in words, recursively, or as an explicit formula.
- Example: In 3, 6, 9, 12, the rule is add 3 each time.
- As a formula, this can be written as aₙ = 3n.
2. How do you find the rule of an arithmetic sequence?
To find the rule of an arithmetic sequence, calculate the common difference and use the formula aₙ = a₁ + (n − 1)d.
- Step 1: Subtract consecutive terms to find the common difference (d).
- Step 2: Identify the first term a₁.
- Step 3: Substitute into the formula.
- Example: 5, 8, 11, 14 → d = 3, so rule is aₙ = 5 + (n − 1)3.
3. What is the formula for the nth term of a sequence?
The formula for the nth term depends on the type of sequence, such as arithmetic or geometric.
- For an arithmetic sequence: aₙ = a₁ + (n − 1)d.
- For a geometric sequence: aₙ = a₁rⁿ⁻¹.
- Example (arithmetic): 2, 5, 8 → aₙ = 2 + (n − 1)3.
4. How do you find the nth term from a sequence?
To find the nth term, identify the pattern and express it as a formula in terms of n.
- Check if the difference between terms is constant.
- If constant, use aₙ = a₁ + (n − 1)d.
- If multiplied each time, use aₙ = a₁rⁿ⁻¹.
- Example: 4, 7, 10 → d = 3 → aₙ = 4 + (n − 1)3.
5. How do you find the rule of a geometric sequence?
To find the rule of a geometric sequence, determine the common ratio and use aₙ = a₁rⁿ⁻¹.
- Step 1: Divide consecutive terms to find the common ratio (r).
- Step 2: Identify the first term a₁.
- Step 3: Substitute into the formula.
- Example: 3, 6, 12, 24 → r = 2 → aₙ = 3 × 2ⁿ⁻¹.
6. What is the difference between arithmetic and geometric sequences?
The main difference is that arithmetic sequences add a constant number, while geometric sequences multiply by a constant number.
- Arithmetic sequence: common difference (e.g., 2, 5, 8).
- Geometric sequence: common ratio (e.g., 2, 6, 18).
- Formulas: aₙ = a₁ + (n − 1)d vs aₙ = a₁rⁿ⁻¹.
7. How do you find a missing term in a sequence?
To find a missing term, identify the sequence pattern and apply the correct rule.
- Check if the sequence is arithmetic or geometric.
- Find the common difference or ratio.
- Apply the rule to calculate the missing value.
- Example: 7, __, 13 → difference is 3 → missing term is 10.
8. Can you give an example of finding the rule of a sequence?
Yes, you can find the rule by identifying the pattern and expressing it algebraically.
- Example: 1, 4, 9, 16.
- These are square numbers: 1², 2², 3², 4².
- The rule is aₙ = n².
- This is a quadratic sequence.
9. What is a recursive rule in sequences?
A recursive rule defines each term using the previous term in the sequence.
- It requires the first term a₁.
- Example: 5, 8, 11, 14.
- Recursive rule: a₁ = 5, aₙ = aₙ₋₁ + 3.
- Each term depends on the one before it.
10. What are common mistakes when finding the rule of a sequence?
Common mistakes include misidentifying the pattern or using the wrong formula for the sequence type.
- Confusing common difference with common ratio.
- Forgetting the (n − 1) in aₙ = a₁ + (n − 1)d.
- Assuming every sequence is arithmetic.
- Not checking the pattern carefully before forming the nth term.
