Sequence Rule Finder
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 Sequences - Finding a Rule
1. What is a Sequence?
Ans: A sequence is an ordered set with elements known as terms. Usually, terms in a sequence are numbers and a sequence can have an infinite number of terms. An example of a sequence is 2, 4, 6, 8, 10, 12… Here the first term is 2, the second term is 4, the third term is 6, and so on. In Mathematics, there are four types of sequence namely: arithmetic Sequence, geometric sequence, Fibonacci Sequence, and harmonic sequence.
2. What is an Arithmetic Sequence?
Ans: An arithmetic sequence is a sequence where the difference between any two consecutive terms is the same. For example 1, 3, 5, 7, 9, 11, 13, is an arithmetic sequence with first term 1, common difference 2, and the number of terms 7.
3. What are the Benefits of Finding the General Term of a Given Sequence.
Ans: A sequence is an arrangement of a number in a particular order. A sequence has both terms and values. The terms are positions of a number in a given sequence. You might be asked to find the nth term of a sequence. Finding general rules for nth terms of a given sequence is an easy approach to calculate the unknown terms.
