How to Find the General Term of Sequences
What Is a Sequence?
A sequence is a function whose domain is an ordered list of numbers. These numbers are positive integers starting with 1. Sometimes, people mistakenly use the terms series and sequence. A sequence is a set of positive integers while series is the sum of these positive integers. The denotation for the terms in a sequence is:
a_{1}, a_{2}, a_{3}, a_{4}, a_{n}, . . .
Finding the nth term of a sequence is easy given a general equation. But doing it the other way around is a struggle. Finding a general equation for a given sequence requires a lot of thinking and practice but, learning the specific rule guides you in discovering the general equation. In this article, you will learn how to induce the patterns of sequences and write the general term when given the first few terms. There is a stepbystep guide for you to follow and understand the process and provide you with clear and correct computations.
What Is an Arithmetic Sequence?
An arithmetic series is a series of ordered numbers with a constant difference. In an arithmetic sequence, you will observe that each pair of consecutive terms differs by the same amount. For example, here are the first five terms of the series.
3, 8, 13, 18, 23
Do you notice a special pattern? It is obvious that each number after the first is five more than the preceding term. Meaning, the common difference of the sequence is five. Usually, the formula for the nth term of an arithmetic sequence whose first term is a_{1} and whose common difference is d is displayed below.
a_{n} = a_{1} + (n  1) d
Steps in Finding the General Formula of Arithmetic and Geometric Sequences
1. Create a table with headings n and a_{n} where n denotes the set of consecutive positive integers, and a_{n} represents the term corresponding to the positive integers. You may pick only the first five terms of the sequence. For example, tabulate the series 5, 10, 15, 20, 25, . . .
n
 an


1
 5

2
 10

3
 15

4
 20

5
 25

2. Solve the first common difference of a. Consider the solution as a tree diagram. There are two conditions for this step. This process applies only to sequences whose nature are either linear or quadratic.
Condition 1: If the first common difference is a constant, use the linear equation ax + b = 0 in finding the general term of the sequence.
a. Pick two pairs of numbers from the table and form two equations. The value of n from the table corresponds to the x in the linear equation, and the value of a_{n} corresponds to the 0 in the linear equation.
a(n) + b = a_{n}
b. After forming the two equations, calculate a and b using the subtraction method.
c. Substitute a and b to the general term.
d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.
Condition 2: If the first difference is not constant and the second difference is constant, use the quadratic equation ax^{2} + b(x) + c = 0.
a. Pick three pairs of numbers from the table and form three equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.
an^{2} + b(n) + c = a_{n}
b. After forming the three equations, calculate a, b, and c using the subtraction method.
c. Substitute a, b, and c to the general term.
d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.
Problem 1: General Term of an Arithmetic Sequence Using Condition 1
Find the general term of the sequence 7, 9, 11, 13, 15, 17,. . .
Solution
a. Create a table of a_{n} and n values.
n
 an


1
 7

2
 9

3
 11

4
 13

5
 15

6
 17

b. Take the first difference of a_{n}.
c. The constant difference is 2. Since the first difference is a constant, therefore the general term of the given sequence is linear. Pick two sets of values from the table and form two equations.
General Equation:
an + b = a_{n}
Equation 1:
at n = 1, a_{1} = 7
a (1) + b = 7
a + b = 7
Equation 2:
at n = 2 , a_{2} = 9
a (2) + b = 9
2a + b = 9
d. Subtract the two equations.
(2a + b = 9)  (a + b = 7)
a = 2
e. Substitute the value of a = 2 in equation 1.
a + b = 7
2 + b = 7
b = 7  2
b = 5
f. Substitute the values a = 2 and b = 5 in the general equation.
an + b = a_{n}
2n + 5 = a_{n}
g. Check the general term by substituting the values into the equation.
a_{n} = 2n + 5_{}
a_{1}= 2(1) + 5 = 7_{}
a_{2}= 2(2) + 5 = 9
a_{3}= 2(3) + 5 = 11
a_{4}= 2(4) + 5 = 13
a_{5}= 2(5) + 5 = 15
a_{6}= 2(6) + 5 = 17
Therefore, the general term of the sequence is:
a_{n} = 2n + 5
Problem 2: General Term of Arithmetic Sequence Using Condition 2
Find the general term of the sequence 2, 3, 5, 8, 12, 17, 23, 30,. . .
Solution
a. Create a table of a_{n} and n values.
n
 an


1
 2

2
 3

3
 5

4
 8

5
 12

6
 17

7
 23

8
 30

b. Take the first difference of a_{n}. If the first difference of a_{n} is not constant, take the second.
c. The second difference is 1. Since the second difference is a constant, therefore the general term of the given sequence is quadratic. Pick three sets of values from the table and form three equations.
General Equation:
an^{2} + b(n) + c = a_{n}
Equation 1:
at n = 1, a_{1} = 2
a (1) + b (1) + c = 2
a + b + c = 2
Equation 2:
at n = 2, a_{2} = 3
a (2)^{2} + b (2) + c = 3
4a + 2b + c = 3
Equation 3:
at n = 3, a_{2} = 5
a (3)^{2} + b (3) + c = 5
9a + 3b + c = 5
d. Subtract the three equations.
Equation 2  Equation 1: (4a + 2b + c = 3)  (a + b + c = 2)
Equation 2  Equation 1: 3a + b = 1
Equation 3  Equation 2: (9a + 3b + c = 5)  (4a + 2b + c = 3)
Equation 3  Equation 2: 5a + b = 2
(5a + b = 2)  (3a + b = 1)
2a = 1
a = 1/2
e. Substitute the value of a = 1/2 in any of the last two equations.
3a + b = 1
3 (1/2) + b = 1
b = 1  3/2
b =  1/2
a + b + c = 2
1/2  1/2 + c = 2
c = 2
f. Substitute the values a = 1/2, b = 1/2, and c = 2 in the general equation.
an^{2} + b(n) + c = a_{n}
(1/2)n^{2}  (1/2)(n) + 2 = a_{n}
g. Check the general term by substituting the values into the equation.
(1/2)n^{2}  (1/2)(n) + 2 = a_{n}
a_{n =} 1/2 (n^{2}  n + 4)
a_{1} = 1/2 (1^{2}  1 + 4) = 2
a_{2} = 1/2 (2^{2}  2 + 4) = 3
a_{3} = 1/2 (3^{2}  3 + 4) = 5
a_{4} = 1/2 (4^{2}  4 + 4) = 8
a_{5} = 1/2 (5^{2}  5 + 4) = 12
a_{6} = 1/2 (6^{2}  6 + 4) = 17
a_{7} = 1/2 (7^{2}  7 + 4) = 23
Therefore, the general term of the sequence is:
a_{n =} 1/2 (n^{2}  n + 4)
Problem 3: General Term of Arithmetic Sequence Using Condition 2
Find the general term for the sequence 2, 4, 8, 14, 22, . . .
Solution
a. Create a table of a_{n} and n values.
n
 an


1
 2

2
 4

3
 8

4
 14

5
 22

b. Take the first and second difference of a_{n}.
c. The second difference is 2. Since the second difference is a constant, therefore the general term of the given sequence is quadratic. Pick three sets of values from the table and form three equations.
General Equation:
an^{2} + b(n) + c = a_{n}
Equation 1:
at n = 1, a_{1} = 2
a (1) + b (1) + c = 2
a + b + c = 2
Equation 2:
at n = 2, a_{2} = 4
a (2)^{2} + b (2) + c = 4
4a + 2b + c = 4
Equation 3:
at n = 3, a_{2} = 8
a (3)^{2} + b (3) + c = 8
9a + 3b + c = 8
d. Subtract the three equations.
Equation 2  Equation 1: (4a + 2b + c = 4)  (a + b + c = 2)
Equation 2  Equation 1: 3a + b = 2
Equation 3  Equation 2: (9a + 3b + c = 8)  (4a + 2b + c = 4)
Equation 3  Equation 2: 5a + b = 4
(5a + b = 4)  (3a + b = 2)
2a = 2
a = 1
e. Substitute the value of a = 1 in any of the last two equations.
3a + b = 2
3 (1) + b = 2
b = 2  3
b =  1
a + b + c = 2
1  1 + c = 2
c = 2
f. Substitute the values a = 1, b = 1, and c = 2 in the general equation.
an^{2} + b(n) + c = a_{n}
(1)n^{2}  (1)(n) + 2 = a_{n}
n^{2}  n + 2 = a_{n}
g. Check the general term by substituting the values into the equation.
n^{2}  n + 2 = a_{n}
a_{1 =} 1^{2}  1 + 2 = 2
a_{2 =} 2^{2}  2 + 2 = 4
a_{3 =} 3^{2}  3 + 2 = 8
a_{4 =} 4^{2}  4 + 2 = 14
a_{5 =} 5^{2}  5 + 2 = 22
Therefore, the general term of the sequence is:
a_{n =} n^{2}  n + 2
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© 2018 John Ray
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