Ray is a Licensed Engineer in the Philippines. He loves to write any topic about mathematics and civil engineering.

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:

a₁, a₂, a₃, a₄, 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 step-by-step 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 a₁ and whose common difference is d is displayed below.

a_n = a₁ + (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, . . .

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 ax² + 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.

an² + 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.

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, a₁ = 7

a (1) + b = 7

a + b = 7

Equation 2:

at n = 2 , a₂ = 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

a₁= 2(1) + 5 = 7

a₂= 2(2) + 5 = 9

a₃= 2(3) + 5 = 11

a₄= 2(4) + 5 = 13

a₅= 2(5) + 5 = 15

a₆= 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.

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:

an² + b(n) + c = a_n

Equation 1:

at n = 1, a₁ = 2

a (1) + b (1) + c = 2

a + b + c = 2

Equation 2:

at n = 2, a₂ = 3

a (2)² + b (2) + c = 3

4a + 2b + c = 3

Equation 3:

at n = 3, a₂ = 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.

an² + b(n) + c = a_n

(1/2)n² - (1/2)(n) + 2 = a_n

g. Check the general term by substituting the values into the equation.

(1/2)n² - (1/2)(n) + 2 = a_n

a_{n =} 1/2 (n² - n + 4)

a₁ = 1/2 (1² - 1 + 4) = 2

a₂ = 1/2 (2² - 2 + 4) = 3

a₃ = 1/2 (3² - 3 + 4) = 5

a₄ = 1/2 (4² - 4 + 4) = 8

a₅ = 1/2 (5² - 5 + 4) = 12

a₆ = 1/2 (6² - 6 + 4) = 17

a₇ = 1/2 (7² - 7 + 4) = 23

Therefore, the general term of the sequence is:

a_{n =} 1/2 (n² - 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.

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:

an² + b(n) + c = a_n

Equation 1:

at n = 1, a₁ = 2

a (1) + b (1) + c = 2

a + b + c = 2

Equation 2:

at n = 2, a₂ = 4

a (2)² + b (2) + c = 4

4a + 2b + c = 4

Equation 3:

at n = 3, a₂ = 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.

an² + b(n) + c = a_n

(1)n² - (1)(n) + 2 = a_n

n² - n + 2 = a_n

g. Check the general term by substituting the values into the equation.

n² - n + 2 = a_n

a_{1 =} 1² - 1 + 2 = 2

a_{2 =} 2² - 2 + 2 = 4

a_{3 =} 3² - 3 + 2 = 8

a_{4 =} 4² - 4 + 2 = 14

a_{5 =} 5² - 5 + 2 = 22

Therefore, the general term of the sequence is:

a_{n =} n² - n + 2

Self-Assessment

For each question, choose the best answer. The answer key is below.

1. Find the general term of the sequence 25, 50, 75, 100, 125, 150, ...
   • an = n + 25
   • an = 25n
   • an = 25n^2
2. Find the general term of the sequence 7/2, 13/2, 19/2, 25/2, 31/2,...
   • an = 3 + n/2
   • an = n + 3/2
   • an = 3n + 1/2

Answer Key

1. an = 25n
2. an = 3n + 1/2

Interpreting Your Score

If you got 0 correct answers: Sorry, try again!

If you got 2 correct answers: Good Job!

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Questions & Answers

Question: How to find general term of sequence 0, 3, 8, 15, 24?

Answer: The general term for the sequence is an = a(n-1) + 2(n+1) + 1

Question: whats is the general term of the set {1,4,9,16,25}?

Answer: The general term of the sequence {1,4,9,16,25} is n^2.

Question: How do I get the formula if the common difference falls on the third row?

Answer: If the constant difference falls on the third, the equation is a cubic. Try solving it following the pattern for quadratic equations. If it's not applicable, you can solve it using logic and some trial and error.

Question: How to find general term of the sequence 4, 12, 26, 72, 104, 142, 186?

Answer: The general term of the sequence is an = 3n^2 − n + 2. The sequence is quadratic with second difference 6. The general term has the form an = αn^2+βn+γ.To find α, β, γ plug in values for n = 1, 2, 3:

4 = α + β + γ

12 = 4α + 2β + γ

26 = 9α + 3β + γ

and solve, yielding α = 3, β = −1, γ = 2

Question: What is the general term of sequence 6,1,-4,-9?

Answer: This is a simple arithmetic sequence. It follows the formula an = a1 + d(n-1). But in this case, the second term has to be negative an = a1 - d(n-1).

At n = 1, 6 - 5(1-1) = 6

At n = 2, 6 - 5(2-1) = 1

At n = 3, 6 - 5(3-1) = -4

At n = 4, 6 - 5(4-1) = -9

Question: What will be nth term of the sequence 4, 12, 28, 46, 72, 104, 142 ...?

Answer: Unfortunately, this sequence does not exist. But if you replace 28 with 26. The general term of the sequence would be an = 3n^2 − n + 2

Question: How to find the general term for the sequence 1/2 , 2/3 ,3/4 ,4/5...?

Answer: For the given sequence the general term could be defined as n/(n + 1), where ’n’ is clearly a natural number.

Question: Is there a faster way to calculate the general term of a sequence?

Answer: Unfortunately, this is the easiest method in finding the general term of basic sequences. You can refer to your textbooks or wait until I get to write another article regarding your concern.

Question: What is the explicit formula for the nth term of the sequence 1,0,1,0?

Answer: The explicit formula for the nth term of the sequence 1,0,1,0 is an = 1/2 + 1/2 (−1)^n, wherein the index starts at 0.

Question: What is the set builder notation of an empty set?

Answer: The notation for an empty set is "Ø."

Question: What is the general formula of the sequence 3,6,12, 24..?

Answer: The general term of the given sequence is an = 3^r^(n-1).

Question: What if there is no common difference for all the rows?

Answer: if there is no common difference for all the rows, try to identify the flow of the sequence through trial and error method. You must identify the pattern first before concluding an equation.

Question: What is the general form of the sequence 5,9,13,17,21,25,29,33?

Answer: The general term of the sequence is 4n + 1.

Question: Is there another way of finding general term of sequences using condition 2?

Answer: There are a lot of ways in solving the general term of sequences, one is trial and error. The basic thing to do is writing down their commonalities and derive equations from those.

Question: How do i find the general term of a sequence 9,9,7,3?

Answer: If this is the correct sequence, the only pattern I see is when you start with number 9.

9

9 - 0 = 9

9 - 2 = 7

9 - 6 = 3

Therefore.. 9 - (n(n-1)) where n starts with 1.

If not, I believe there is a mistake with the sequence you provided. Please try to recheck it.

Question: How to find an expression for the general term of a series 1+1•3+1•3•5+1•3•5•7+...?

Answer: The general term of the series is (2n-1)!.

Question: General term for the sequence {1,4,13,40,121} ?

Answer: 1

1+3 = 4

1+3+3^2 = 13

1+3+3^2+3^3 = 40

1+3+3^2+3^3+3^4 = 121

So, the general term of the sequence is a(sub)n=a(sub)n-1 + 3^(n-1)

Question: How to find general term for sequence given as an=3+4a(n-1) given a1=4?

Answer: So you mean how to find the sequence given the general term. Given the general term, just start substituting the value of a1 in the equation and let n =1. Do this for a2 where n=2 and so on and so forth.

Question: How to find general pattern of 3/7, 5/10, 7/13,...?

Answer: For fractions, you can separately analyze the pattern in the numerator and the denominator.

For the numerator, we can see that the pattern is by adding 2.

3

3 + 2 = 5

5 + 2 = 7

or by adding multiples of 2

3

3 + 2 = 5

3 + 4 = 7

Therefore the general term for the numerator is 2n + 1.

For the denominator, we can observe that the pattern is by adding 3.

7

7 + 3 = 10

10 + 3 = 13

Or by adding multiples of 3

7

7 + 3 = 10

7 + 6 = 13

Therefore, the pattern for the denominator is 3n + 4.

Combine the two patterns and you'll come up with (2n + 1) / (3n + 4) which is the final answer.

Question: What is the general term of the sequence {7,3,-1,-5}?

Answer: The pattern for the given sequence is:

7

7 - 4 = 3

3 - 4 = -1

-1 - 4 = -5

All succeeding terms are subtracted by 4.

Question: How to find the general term of the sequence 8,13,18,23,...?

Answer: First thing to do is try to find a common difference.

13 - 8 = 5

18 - 13 = 5

23 - 18 = 5

Therefore the common difference is 5. The sequence is done by adding 5 to the previous term. Recall that the formula for the arithmetic progression is an = a1 + (n - 1)d. Given a1 = 8 and d = 5, substitute the values to the general formula.

an = a1 + (n - 1)d

an = 8 + (n - 1) (5)

an = 8 + 5n - 5

an = 3 + 5n

Therefore, the general term of the arithmetic sequence is an = 3 + 5n

Question: How to find general term of sequence of -1, 1, 5, 9, 11?

Answer: I actually don't get the sequence really well. But my instinct says it goes like this..

-1 + 2 = 1

1 + 4 = 5

5 +4 = 9

9 + 2 = 11

+2, +4, +4, +2, +4, +4, +2, +4, +4

Question: How to find the general term of 32,16,8,4,2,...?

Answer: I believe each term (except the first term) is found by dividing the previous term by 2.

Question: How to find general term of sequence 1/2, 1/3, 1/4, 1/5?

Answer: You can observe that the only changing portion is the denominator. So, we can set the numerator as 1. Then the common difference of the denominator is 1. So, the expression is n+1.

The general term of the sequence is 1/(n+1)

Question: How to find general term of the sequence 1,6,15,28?

Answer: The general term of the sequence is n(2n-1).

Question: How to find the general term of the sequence 1, 5, 12, 22 ?

Answer: The general term of the sequence 1, 5, 12, 22 is [n(3n-1)]/2.

© 2018 Ray

Comments

Ray (author) from Philippines on April 24, 2020:

Hi, Earnest. I am currently working on my new blogs to be featured on my own website. It'll include topics related to geometric series, geometric sequences, arithmetic series, and arithmetic sequences. There'll be also topics about harmonic series and geometric infinite series. Please send me your e-mail so I can send you an e-mail as soon as my website is up. You may visit my profile to check on my social media and e-mail. Thanks!

Earnest on April 24, 2020:

... some GPs please and also sequences of this kind .1/27+2/37+3/47+4/47+...and Any personal technique could help so much. Thanks for the service!

Ray (author) from Philippines on March 08, 2020:

For those who want to learn finding the nth term of arithmetic sequences and geometric sequences, please leave your comments here in the comment section. Thank you!

Ray (author) from Philippines on January 04, 2020:

Technically, the sequences you provided are not basic sequences and requires an advanced knowledge in the field of mathematics. For instance, the general term of the sequence 1,3,11,43… is n! multiplied by the summation of 1/(n-k) where k starts with 0. Soon, I’ll create an article discussing about these type of sequences.

Akinwunmi Sulaiman Awwal on December 16, 2019:

how to get general term for the sequence

3, 10, 22, 35, 49,

2, 10, 25, 54,

2, 5, 11, 23, 38,

1, 3, 11, 43,

1, 2, 5, 15, 54,

1, 4, 13, 47, 185,

Bitaniya bahiru on October 16, 2019:

Thank you so much. it's very helpful

syed ashar ali on September 18, 2019:

I want to find a generalized term of a sequence whose common ratio is an arithmetic sequence

Nasibud Din on August 25, 2019:

I am not understand dicemal and binary system

godfrey sebabatso mokatile on August 18, 2019:

the maths has no problem is need to practice it

Brian.sitton@ciy.com on June 30, 2019:

I think in the first picture the -1 should be a +1.

(in the +7 sequence)

Jonh paul Sallaya on June 12, 2019:

General term

An.sqaure. + bn + c = An

A=1 ,b=1, c= -1

Then whats next pls?

Han on March 18, 2019:

7,13,25,49……

7=6+1=6x2^0+1

13=6x2+1=6x2^1+1

25=6x4+1=6x2^2+1

49=6x8+1=6x2^3+1……So:

an=6x2^(n-1)+1

Cody on December 29, 2018:

Then what is the sequence for 7,13,25,49.........

lol on November 02, 2018:

this really helped me, thanks

Rowan on October 21, 2018:

That arithmetic sequence is super wrong. -6 + 7 is not -1

Srijan Guha on August 27, 2018:

Easily understandable and very helpful mathematical article.

Nicole Ann Cuevas on July 21, 2018:

Thanks! Its a big help!

Ray (author) from Philippines on June 29, 2018:

Thank you, Ma'am Chitrangada Sharan. I really like to help students having trouble with their Mathematics subject. Have a great day!

Chitrangada Sharan from New Delhi, India on June 29, 2018:

Your article is educative and informative! Very well explained and useful for many.

Thanks for sharing this !