Introduction to Using Algebra Sequences
You can be moved to tears by numbers— provided they are encoded and decoded fast enough.
—Richard Dawkins (1941– ) British Scientist
In this lesson, you'll learn how to use algebra to find the nth term of arithmetic and geometric sequences, and how to find the exact values of terms within arithmetic and geometric sequences that contain algebraic terms.
A sequence is a set of numbers in which each number is generated according to a rule. We can use algebra to help us find the rule of a sequence or a certain term in a sequence. It might be easy to find the next term in a sequence once we have found the rule of the sequence, but what if we knew only a few numbers and wanted to find the hundredth term of the sequence? We wouldn't want to list all 100 terms!
Arithmetic Sequences
An arithmetic sequence is a sequence in which the difference between any term and the term that precedes it is always the same. For instance, the sequence 3, 7, 11, 15, 19,…is an arithmetic sequence. We can find the rule of an arithmetic sequence by taking the difference between any two consecutive terms. In this sequence, the difference between consecutive terms is 4. The rule is +4, so the next term after 19 is 19 + 4, which is 23.
The numbers in a sequence are called terms. The first term of the sequence 3, 7, 11, 15, 19,…is 3, so we say that t_{1} = 3. The second term in the sequence is represented as t_{2} and the third as t_{3}, and we say that the nth term is t_{n}. The n represents an unknown place in the sequence. We use t_{n} to make a general statement about a sequence.
We found the sixth term, t_{6}, of the sequence by adding 4, the difference between any pair of consecutive terms, to the previous term, t_{5}. If we know the first term in an arithmetic sequence and the difference between consecutive terms, we can find any term in the sequence using this formula: t_{n} = t_{1} + (n – 1)d. In this formula, d represents the difference between any term and its previous term, and n represents the place of the term in the sequence.
We already know that the fifth term in the sequence is 19, but let's test the formula to be sure that it works. The first term, t_{1}, is 3. The difference, d, is 4. We are looking for the fifth term, n = 5:
t_{n} = t_{1} + (n – 1)d 
t_{5} = 3 + (5 – 1)4 
t_{5} = 3 + (4)4 
t_{5} = 3 + 16 
t_{5} = 19 
What if the terms of a sequence are algebraic? Use two consecutive terms from the sequence and find their difference. Then, find the nth term of the sequence in terms of the variables used in the sequence.
Example 
x + 2, 3x, 4x + 3, 6x + 1,… 
We can find the fifth term, sixth term, or any other term using the formula t_{n} = t_{1} + (n – 1)d. Let's find the seventh term. 
First, find the difference between any pair of consecutive terms. Subtract the first term from the second term: 3x – (x + 2) = 2x – 2, so d = 2x – 2. The first term, t_{1}, is x + 2. Because we are looking for the seventh term, n = 7: 
t_{7} = (x + 2) + (7 – 1)(2x – 2) 
t_{7} = (x + 2) + (6)(2x – 2) 
t_{7} = (x + 2) + (12x – 12) 
t_{7} = 13x – 10 
The seventh term of the sequence is 13x – 10. We can find the exact value of that term, because we know that the difference between any two consecutive terms in an arithmetic sequence is always the same. That's why it didn't matter which pair of consecutive terms we chose to subtract in order to find d. We found that the difference between the first term and the second term was 2x – 2. The difference between the second term and the third term is (4x + 3x) – 3x = x + 3. Because the difference between terms is always the same, 2x – 2 must equal x + 3. Set these two differences equal to each other and solve for x:
2x – 2 = x + 3 
x – 2 = 3 
x = 5 
The value of x in this sequence is 5, which means that the seventh term, 13x – 10, is equal to 13(5) – 10 = 65 – 10 = 55.
Tip:When finding two different expressions that represent the difference, be sure to subtract terms in the order they appear in the sequence. The difference between the first two terms is the second term minus the first term, and the difference between the third term and the second term is the third term minus the second term. 

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