Skip to content# Nth Term of a Linear Sequence | How to Find It with Examples

## Nth Term of a Linear Sequence

## What Is a Linear Sequence?

## General Formula for the Nth Term

## Steps to Find the Nth Term

## Summary of Steps

## Conclusion

## Practice Questions and Answers on Nth Term of a Linear Sequence

## Solutions

In this article, we will discuss how to find the nth term of a linear sequence, also known as an arithmetic sequence.

This fundamental concept in mathematics allows you to determine any term in a sequence without listing all the preceding terms.

We will discuss are:

- How to find the nth term of a linear sequence
- Solving various problems related to sequences and series

*They are very important in practicing questions for coordinate geometry as well.*

Here is one more link to practice a few extra questions: Maths Genie Nth Term of a Linear Sequence Questions

- A
**linear sequence**is a list of numbers where the difference between any two consecutive terms is always the same. - This constant difference is known as the
**common difference.**

**Example of a Linear Sequence:**

Consider the sequence: **2, 5, 8, 11, 14**

From 2 to 5: **Add 3**

From 5 to 8: **Add 3**

From 8 to 11: **Add 3**

From 11 to 14: **Add 3 **

**Common Difference (A):** 3

The general formula to find the nth term of a linear sequence is:

**nth term = A × n + B**

Where:

**A**is the**common difference****B**is a**constant**(the adjustment needed to match the first term)**n**is the**term number**

Our goal is to find the values of **A** and **B** to create a formula that can calculate any term in the sequence.

To find the general term of a sequence, follow these two main steps:

**Step 1: Find the Common Difference (A) **

**Subtract any term from the term that follows it.**- Ensure the difference is consistent throughout the sequence.

**Step 2: Find the Constant Term (B) **

**Use the first term of the sequence.****Subtract the common difference from the first term: B = First term − A**

** **

** **

**Solved Example 1**

**Question: Here are the first 5 terms of an arithmetic sequence. **

**2, 5, 8, 11, 14**

**Find an expression, in term of n, for the nth term of sequence and using that find 10th term of the sequence.**

**Solution: **

**Given Sequence:** 2, 5, 8, 11, 14

**Step 1: Find the Common Difference (A) **

5 – 2 = **3 **

8 – 5 = **3 **

• **Common difference A = 3 **

**Step 2: Find the Constant Term (B)**

• First term is **2**

B = 2 – 3

= **−1 **

**Step 3: **Nth Term Formula:

**nth term = A × n + B**

**nth term = 3n − 1 **

**Finding a Specific Term **

To find the 10th term:

**nth term = 3 × 10 − 1 **

** nth term = 29**

** **

** **

**Solved Example 2**

**Question: Here are the first 5 terms of an arithmetic sequence. **

**6, 11, 16, 21, 26**

**Find an expression, in term of n, for the nth term of sequence and using that find 52nd term of the sequence.**

**Solution: **

**Given Sequence:** 6, 11, 16, 21, 26

**Step 1: Find the Common Difference (A) **

11 – 6 = **5**

16 – 11 = **5**

• **Common difference A = 5**

**Step 2: Find the Constant Term (B)**

• First term is **6**

B = 6 – 5

= **1 **

**Step 3: **Nth Term Formula:

**nth term = A × n + B**

**nth term = 5n + 1 **

**Finding a Specific Term **

To find the 52nd term:

**nth term = 5 × 52 + 1 **

** nth term = 261**

** **

** **

**Solved Example 3**

**Question: Here are the first 5 terms of a number sequence. **

**3, 7, 11, 15, 19 **

**Write down an expression, in terms of n, for the nth term of this sequence and**** verify whether 319 is a term in the sequence. You must justify your answer.**

**Solution: **

**Given Sequence:** 3, 7, 11, 15, 19

**Step 1: Find the Common Difference (A) **

7 – 3 = **4**

11 – 7 = **4**

• **Common difference A = 4**

**Step 2: Find the Constant Term (B)**

• First term is **3**

B = 3 – 4

= **-1 **

**Step 3: **Nth Term Formula:

**nth term = A × n + B**

nth term = 4n – 1

**To check if 319 is a term in this sequence:**

**Set Up the Equation**

4n − 1 = 319

**Solve for n **

- Add 1 to both sides:

4n = 320

- Divide both sides by 4:

n = 80

**Conclusion **

- Since n = 80 is a whole number, 319 is the 80th term of the sequence.
- Therefore, 319 is a term in the sequence.

**Note: If n had not been a whole number (e.g., n = 80.5), then 319 would not be a term in the sequence.**

**1.** Find the Common Difference (A):

- Subtract consecutive terms to find A.

**2.** Find the Constant Term (B):

- Subtract the common difference from the first term:

**B = First term − A**

**3.** Write the Nth Term Formula:

- Combine A and B into the formula:

**nth term = A × n + B**

**4.** Find Any Term in the Sequence:

- Substitute the desired term number (n) into the formula.

Finding the **nth term of a linear sequence** is a straightforward process once you understand the steps involved:

**1. Determine the common difference** between the terms.

**2. Calculate the constant term** by adjusting the first term.

**3. Formulate the nth term** using the general formula.

**4. Apply the formula** to find any term in the sequence or verify if a number is part of the sequence.

**Question 1: Sequence:** 4, 9, 14, 19, 24

**Tasks:**- Find the
**nth term formula.** - Calculate the
**30th term.**

- Find the

**Question 2: Sequence: **15, 12, 9, 6, 3

**Tasks:**- Find the nth term formula.
- Determine if −12 is a term in the sequence.

**Question 3: Sequence:** −5, 0, 5, 10, 15

**Tasks:**- Find the
**nth term formula.** - Find the
**25th term.**

- Find the

**Question 4: Sequence:** 7, 10, 13, 16, 19

**Tasks:**- Find the
**nth term formula.** - Calculate the
**15th term.**

- Find the

**Question 5: Sequence:** 20, 17, 14, 11, 8

**Tasks:**- Find the
**nth term formula.** - Determine if
**−22**is a term in the sequence.

- Find the

**Question 6: Sequence:** 1, 4, 7, 10, 13

**Tasks:**- Find the
**nth term formula.** - Find the
**100th term.**

- Find the

**Question 7:** **Sequence:** −2, 0, 2, 4, 6

**Tasks:**- Find the
**nth term formula.** - Calculate the
**50th term.**

- Find the

**Question 8: Sequence:** 100, 95, 90, 85, 80

**Tasks:**- Find the
**nth term formula.** - Determine if
**50**is a term in the sequence.

- Find the

**Question 9: Sequence:** 5, 9, 13, 17, 21

**Tasks:**- Find the
**nth term formula.** - Find the
**40th term.**

- Find the

**Question 10: Sequence:** 50, 47, 44, 41, 38

**Tasks:**- Find the
**nth term formula.** - Determine if
**2**is a term in the sequence.

- Find the

**Question 1: **

**Step 1:** Find the Common Difference (A)

9 – 4 = 5

**A = 5 **

**Step 2:** Find the Constant Term (B)

- First term is
**4**

B = 4 – 5

**= −1 **

**Nth Term Formula **

nth term = 5n − 1

**Calculate the 30th Term **

nth term = 5 × 30 − 1

= **149**

**Question 2:**

**Step 1:** Find the Common Difference (A)

12 – 15 = −3

**A = −3 **

**Step 2:** Find the Constant Term (B)

- First term is 15

B = 15 – (−3)

**= 18 **

**Nth Term Formula **

nth term = −3n + 18

** Determine if −12 Is in the Sequence **

Set up the equation:

−3n + 18 = **−12 **

**Solve for n: **

1. Subtract 18 from both sides:

−3n = −30

2. Divide both sides by −3:

**n = 10 **

**Conclusion **

- Since
**n = 10**is a whole number,**−12**is the 10th term. - Therefore,
**−12 is a term in the sequence.**

**Question 3:**

**Step 1:** Find the Common Difference (A)

0 – (−5) = 5

**A = 5 **

**Step 2:** Find the Constant Term (B)

- First term is
**−5**

B = (−5) – 5 = **−10 **

**Nth Term Formula **

nth term = 5n − 10

**Find the 25th Term **

nth term = 5 × 25 − 10

**= 115**

**Question 4:**

**Step 1:** Find the Common Difference (A)

10 – 7 = 3

**A = 3 **

**Step 2:** Find the Constant Term (B)

- First term is
**7**

B = 7 – 3 = **4 **

**Nth Term Formula **

nth term = 3n + 4

**Calculate the 15th Term **

nth term = 3 × 15 + 4

**= 49**

**Question 5:**

**Step 1:** Find the Common Difference (A)

17 – 20 = −3

**A = −3 **

**Step 2:** Find the Constant Term (B)

- First term is
**20**

B = 20 – (−3) = **23 **

**Nth Term Formula **

nth term = −3n + 23

**Determine if −22 Is in the Sequence **

Set up the equation:

−3n + 23 = −22

**Solve for n: **

1. Subtract 23 from both sides:

−3n = −45

2. Divide both sides by −3:

** n = 15 **

**Conclusion **

- Since
**n = 15**is a whole number,**−22**is the 15th term. - Therefore,
**−22 is a term in the sequence.**

**Question 6:**

**Step 1:** Find the Common Difference (A)

4 – 1 = 3

**A = 3 **

**Step 2:** Find the Constant Term (B)

- First term is
**1**

B = 1 – 3 = **−2 **

**Nth Term Formula **

nth term = 3n − 2

**Find the 100th Term **

nth term = 3 × 100 − 2

**= 298**

**Question 7:**

**Step 1:** Find the Common Difference (A)

• 0 minus (−2) equals 2

• A = 2

**Step 2:** Find the Constant Term (B)

- First term is
**−2**

B = (−2) – 2 = **−4 **

**Nth Term Formula**

nth term = 2n − 4

**Calculate the 50th Term **

nth term = 2 × 50 − 4

**= 96**

**Question 8:**

**Step 1:** Find the Common Difference (A)

95 – 100 = −5

**A = −5 **

**Step 2:** Find the Constant Term (B)

- First term is
**100**

B = 100 – (−5) = **105 **

**Nth Term Formula **

nth term = −5n + 105

**Determine if 50 Is in the Sequence **

Set up the equation:

−5n + 105 = 50

**Solve for n: **

1. Subtract 105 from both sides:

−5n = −55

2. Divide both sides by −5:

**n = 11 **

**Conclusion **

- Since
**n = 11**is a whole number,**50**is the 11th term. - Therefore,
**50 is a term in the sequence.**

**Question 9:**

**Step 1:** Find the Common Difference (A)

9 – 5 = 4

**A = 4 **

**Step 2:** Find the Constant Term (B)

- First term is
**5**

B = 5 – 4 = **1 **

**Nth Term Formula **

nth term = 4n + 1

**Find the 40th Term **

nth term = 4 × 40 + 1

**= 161**

**Question 10:**

**Step 1:** Find the Common Difference (A)

47 – 50 = −3

**A = −3**

**Step 2:** Find the Constant Term (B)

- First term is
**50**

B = 50 – (−3) = **53 **

**Nth Term Formula **

nth term = −3n + 53

**Determine if 2 Is in the Sequence **

Set up the equation:

−3n + 53 = 2

**Solve for n: **

1. Subtract 53 from both sides:

−3n = −51

2. Divide both sides by −3:

**n = 17 **

**Conclusion **

- Since
**n = 17**is a whole number,**2**is the 17th term. - Therefore,
**2 is a term in the sequence.**