# \({n^{th}}\) Term of an AP

Johan Carl Fredrich Gauss, the father of arithmetic progressions, was asked to find the sum of integers from 1 to 100 without using a counting frame.

This was unheard of, but Gauss, the genius that he was, took up the challenge.

He listed the first 50 integers, and wrote the subsequent 50 in reverse order below the first set.

To his surprise, the sums of the numbers next to each other was 101 i.e. 100 + 1, 99 + 2, 51 + 50, etc.

He found there were 50 such pairs and ended up multiplying 101 with 50 to give an output 5050

Does this confuse you like it has confused Jack?

Stay tuned to learn more about nth term of arithmetic progression.

**Lesson Plan**

**What Is Meant by Arithmetic Progression?**

Arithmetic progression can be defined as a sequence where the differences between every two consecutive terms are the same.

Consider the following AP:

2, 5, 8, 11, 14

The first term a of this AP is 2, the second term is 5, the third term is 8, and so on. We write this as follows:

T_{1} = a = 2

T_{2} = 5

T_{3} = 8

The nth term of this AP will be denoted by T_{n}.

For example, what will be the value of the following terms?

T_{20}, T_{45}, T_{90}, T_{200}

- First term, as the name suggests, the first term of an AP is the first number of the progression. It is usually represented by a.
- As arithmetic progression is a sequence where each term, except the first term, is obtained by adding a fixed number to its previous term, here, the “fixed number” is called the “common difference” and is denoted by d.
- The nth term of arithmetic progression depends on the first term and the common difference of the arithmetic progression.

**How to Determine the Nth Term of AP ?**

We cannot evaluate each and every term of the AP to determine these specific terms. Instead, we must develop a relationship that enables us to find the nth term for any value of n.

To do that, consider the following relations for the terms in an AP:

T_{1} = a

T_{2} = a + d

T_{3} = a + d + d = a + 2d

T_{4} = a + 2d + d = a + 3d

T_{5} = a + 3d + d = a + 4d

T_{6} = a + 4d + d = a + 5d

What pattern do you observe?

If we have to calculate the sixth term, for example, then we have to add five times d (common difference) to the first term a. Similarly, if we have to calculate the n^{th} term, how many times will we add d to a?

The answer should be easy: one less than n.

Thus, the formula of nth term of ap is,

T_{n} = a + (n - 1)d

This relation helps us calculate any term of an AP, given its first term and its common difference.

Thus, for the AP above, we have:

T_{20} = 2 + (20 - 1) 3 = 2 + 57 = 59

T_{45} = 2 + (45 - 1) 3 = 2 + 132 = 134

T_{90} = 2 + (90 - 1) 3 = 2 + 267 = 269

T_{200} = 2 + (200 - 1) 3 = 2 + 597 = 599

**Examples**

**Example 1:** What is the 11th term for the given arithmetic progression?

2, 6, 10, 14, 18,....

Solution:

In the given arithmetic progression,

First term = a = 2

Common difference = d = 4

Term to be found, n = 11

Hence, the 11th term for the given progression is,

T_{n} = a + (n - 1)d

T_{11} = 2 + (11 - 1)4 = 2 + 40 = 42

\(\therefore\) 11th term of AP is 42 |

**Example 2: **If the 5th term of an AP is 40 with a common difference of 6. Find out the arithmetic progression.

Solution:

The given values for the AP are,

Fifth term = T_{5} = 40

Common difference = d = 6

Hence, the fifth term can be written as,

T_{5} = a + (5 - 1)6 = a + 24 = 40

\(\implies\) a = 40 - 24 = 16

Hence, the arithmetic progression is,

T_{1} = a = 16

T_{2} = a + d = 16 + 6 = 22

T_{3} = a + 2d = 16 + (2)(6) = 28

T_{4} = a + 3d = 16 + (3)(6) = 34

T_{5} = a + 4d = 16 + (4)(6) = 40

T_{6} = a + 5d = 16 + (5)(6) = 46

The arithmetic progression is, 16, 22, 28, 34, 40, 46, and so on.

\(\therefore\) AP is 16, 22, 28, 34, 40, 46, and so on |

- The common difference doesn't need to be positive always.
- The common difference of an AP is the difference between two consecutive terms of the AP taken in this order.
- As per the formula of nth term of ap, the first term can be found by using a = T
_{n}- (n - 1)d.

**Solved Examples**

Example 1 |

How can Justin find the 20th term of an AP whose 3rd term is 5 and 7th term is 13?

**Solution**

From the given problem Justin can find nth term of ap, where n = 20 in the following way:

He knows as per the nth term of ap formula,

T_{3} = a + 2d = 5

T_{7} = a + 6d = 13

\(\implies\) 4d = 8

\(\implies\) d = 2

As the 3rd term is 5, the value of a can be given as,

a + (2)2 = 5

\(\implies\) a = 1

Now the term can be calculated as,

T_{20} = a + 19d = 1 + 19(2) = 39

\(\therefore\) The 20th term of AP is 39. |

Example 2 |

Help Jack determine how many three-digit numbers are divisible by 3?

**Solution**

Jack knows the smallest three-digit number which is divisible by 3 is 102 and the largest three-digit number divisible by 3 is 999.

To find the number of terms in the following AP:

102, 105, 108,..,999

He will take 999 be the nth term of AP, it can be seen that a is equal to 102, and d is equal to 3.

Thus as per nth term of ap formula,

T_{n} = a + (n - 1)d = 102 + (n - 1)3 = 999

\(\implies\) 3(n - 1) = 999 - 102 = 897

\(\implies\) n - 1 = \(\dfrac{897}{3}\)

\(\implies\) n = 300

\(\therefore\) There are 300 three-digit numbers which are divisible by 3 |

Example 3 |

Maria considered the below AP:

7, 11, 15, 19,...

How will she determine if the number 301 a part of this AP?

**Solution**

Maria knows a is equal to 7 and d is equal to 4.

To determine if 301 is a part of AP or not,

Maria will consider 301 be the nth term of this AP, where n is a positive integer.

As per nth term of ap formula,

T_{n} = a + (n - 1)d = 7 + (n - 1)4 = 301

\(\implies\) 4(n - 1) = 301 - 7 = 294

\(\implies\) n - 1 = \(\dfrac{294}{4} = \dfrac{147}{2}\)

\(\implies\) n = \(\dfrac{149}{2}\)

Maria obtained n as a non-integer, whereas n should have been an integer.

This can only mean that 301 is not part of the given AP.

\(\therefore\) 301 is not a part of this AP |

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize **

We hope you enjoyed learning about nth term of arithmetic progression and nth term of ap formula with the practice questions. Now you can find nth term of ap using the formula of nth term of ap.

The mini-lesson targeted the fascinating concept of nth term of arithmetic progression. The math journey around graphing functions starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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**FAQs on Nth term of AP**

### 1. What is AP in maths?

AP is elaborated as arithmetic progression in maths. It is defined as as a sequence where each term, except the first term, is obtained by adding a fixed number to its previous term.

### 2. What is A in AP?

A in AP is elaborated as arithmetic.

### 3. What is the formula for the nth term of an AP?

The formula for the nth term of an AP is, T_{n} = a + (n - 1)d.

### 4.What is the formula of sum of AP?

The formula of sum of AP is, S_{n} = \(\frac{n}{2}\)(2a + (n - 1)d).

### 5. What is the formula for sum of n natural numbers?

The formula for sum of n natural numbers is, \(\frac{n(n + 1)}{2}\)

### 6. Is arithmetic progression infinite?

An arithmetic progression can be either infinite or finite.

### 7. What is finite AP and infinite AP?

- The AP where there are limited number of terms in a sequence, it is known as a finite AP. For example, 2, 4, 6, 8
- The AP where there is no limit on number of terms in a sequence, it is known as an infinite AP. For example, 5, 10, 15, 20,....

### 8. What is non constant arithmetic progression?

The non constant arithmetic progression in defined as a sequence having common differences other than 0. For example, 1, 2, 3, 4 etc.

### 9. How do you find the nth term of a sequence with different differences?

The steps to find the nth term of a sequence with different differences are:

- We take the difference between the consecutive terms.
- If the difference among consecutive terms is not constant, we check the change in difference occurring.
- If the change in difference occurring is a, then the nth term is given as (\(\dfrac{a}{2}\))n
^{2}.

### 10. What is difference between an arithmetic sequence and an arithmetic series?

- An arithmetic sequence is defined as a sequence where the common difference among the successive terms is constant.
- An arithmetic series the sum of all terms of an arithmetic sequence.