Johan Carl Fredrich Gauss, the dad of arithmetic progressions, was asked to find the amount of integers indigenous 1 to 100 without making use of a count frame.

You are watching: How to find the 100th term in a sequence

This was unheard of, yet Gauss, the genius that he was, take it up the challenge.

He noted the an initial 50 integers, and also wrote the subsequent 50 in reverse order below the an initial set.

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

He uncovered there were 50 together pairs and also ended up multiplying 101 through 50 to offer an calculation 5050

Does this confuse you choose it has perplexed Jack?


Stay tuned come learn more aboutnth term of arithmetic progression.

Lesson Plan

1.What Is meant by Arithmetic Progression?
2.Important notes on Nth term of Arithmetic Progression
3.Tips and Tricks
4.Solved examples on Nth hatchet of Arithmetic Progression
5.Interactive inquiries on Nth term of Arithmetic Progression

What Is intended by Arithmetic Progression?

Arithmetic progression have the right to be identified asa sequence whereby the differences in between every two consecutive terms space the same.

Consider the following AP:

2, 5, 8, 11, 14

The very first termaof this AP is 2, the 2nd term is 5, the third term is 8, and also so on. We compose this as follows:

T1= a = 2

T2= 5

T3= 8

Thenthterm of this AP will be denoted byTn.

For example, what will be the worth of the complying with terms?

T20, T45, T90, T200


Important Notes
First term, as the surname suggests, the first term of one AP is the first number the the progression. The is usually stood for by a.As arithmetic development is a sequence where each term, other than the an initial term, is derived by including a addressed number come its previous term, here, the “fixed number” is called the “common difference” and is denoted byd.Thenth hatchet of arithmetic progression depends on the very first term and the usual difference that the arithmetic progression.

How to identify the Nth hatchet of AP?

We cannot evaluate each and every hatchet of the AP to recognize these details terms. Instead, we must develop a relationship that allows us to find thenthterm for any kind of value ofn.

To do that, take into consideration the adhering to relations for the terms in an AP:

T1 = a

T2 = a + d

T3 = a + d + d = a + 2d

T4= a + 2d + d = a + 3d

T5= a + 3d + d = a + 4d

T6= a + 4d + d = a + 5d

What pattern carry out you observe?

If wehave to calculation the sixth term, because that example, then wehave to include five timesd (common difference)to the very first terma. Similarly, if wehave to calculation thenthterm, how plenty of times will weadddtoa?

The answer have to be easy: one less thann.

Thus, the formulaof nth term of ap is,

Tn= a +(n - 1)d

This relation helps united state calculate any type of term of one AP, provided its first term and its typical difference.

Thus, because that the AP above, us have:

T20= 2 + (20 - 1) 3 = 2 + 57 = 59

T45= 2 + (45- 1) 3 = 2 + 132= 134

T90= 2 + (90- 1) 3 = 2 + 267 = 269

T200= 2 + (200 - 1) 3 = 2 + 597 = 599


Example 1:What is the 11th term because that the provided arithmetic progression?

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


In the provided arithmetic progression,

First ax = a = 2

Common distinction = d = 4

Term to be found, n = 11

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

Tn = a + (n - 1)d

T11= 2+ (11 -1)4 = 2 + 40 = 42

\(\therefore\) 11th hatchet of AP is 42

Example 2:If the 5th term of one AP is 40 with a usual difference of 6. Uncover out the arithmetic progression.


The provided values for the AP are,

Fifthterm =T5 = 40

Common difference = d = 6

Hence, the fifth term deserve to be written as,

T5= a+ (5- 1)6= a+ 24= 40

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

Hence, the arithmetic development is,

T1 = a = 16

T2 = a + d = 16 + 6 = 22

T3 =a + 2d = 16 + (2)(6) =28

T4= a + 3d = 16 + (3)(6) = 34

T5=a + 4d =16 + (4)(6) = 40

T6=a + 5d =16 + (5)(6) = 46

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

\(\therefore\) AP is16, 22, 28, 34, 40, 46, and also so on


Example 1

How have the right to Justin uncover the 20th ax of one AP whose 3rd term is 5 and 7th hatchet is 13?


From the given problem Justin can uncover nth hatchet of ap, whereby n = 20 in the following way:

He knows as per the nth term of ap formula,

T3= a + 2d = 5

T7= a + 6d = 13

\(\implies\) 4d = 8

\(\implies\) d = 2

As the 3rd term is 5, the worth of a deserve to be offered as,

a + (2)2= 5

\(\implies\) a= 1

Now the term deserve to be calculation as,

T20= a + 19d = 1 + 19(2) = 39

\(\therefore\) The 20th hatchet of AP is 39.
Example 2

Help Jack determine how countless three-digit numbers room divisible through 3?


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

To uncover the number of terms in the following AP:

102, 105, 108,..,999

He will take 999 be thenthterm of AP, it deserve to be viewed that ais equal to 102, anddis same to 3.

Thus as per nth ax of ap formula,

Tn = a + (n - 1)d = 102 + (n - 1)3 = 999

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

\(\implies\) n - 1 = \(\dfrac8973\)

\(\implies\) n = 300

\(\therefore\) There room 300 three-digit numbers which room divisible by 3
Example 3

Maria thought about the listed below AP:

7, 11, 15, 19,...

How will certainly she recognize ifthe number 301 a part of this AP?


Maria knows ais equal to 7 anddis equal to 4.

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

Maria will consider301 it is in thenthterm the this AP, wherenis a positive integer.

As pernth ax of ap formula,

Tn = a + (n - 1)d = 7+ (n - 1)4= 301

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

\(\implies\) n - 1 = \(\dfrac2944 = \dfrac1472\)

\(\implies\) n = \(\dfrac1492\)

Maria obtainednas a non-integer, whereasnshould have been one integer.

This deserve to only average that 301 is not part of the provided AP.

\(\therefore\) 301 is no a component of this AP

Interactive Questions

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Let's summarize

We expect you took pleasure in learning around nth term of arithmetic progression and also nth ax of ap formula with the exercise questions. Currently you canfind nth ax of ap using the formula of nth term of ap.

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FAQs onNth hatchet of AP

1.What is AP in maths?

AP is elaborated together arithmetic development in maths.It is defined asas a sequence whereby each term, other than the very first term, is derived by including a solved number come its vault term.

2.What is A in AP?

A in AP is elaborated together arithmetic.

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

The formula because that the nth hatchet of one AP is,Tn = a + (n - 1)d.

4.What is the formula of sum of AP?

The formula of sum of AP is,Sn = \(\fracn2\)(2a + (n - 1)d).

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

The formula for amount of n herbal numbers is, \(\fracn(n + 1)2\)

6.Is arithmetic development infinite?

An arithmetic progression deserve to be either boundless orfinite.

7.What is finite AP and infinite AP?

The AP whereby there are limited number that termsin a sequence, it is known as a limited AP. Because that example, 2, 4, 6, 8The AP whereby there is no border on variety of terms in a sequence, the is known as an unlimited AP. Because that example, 5, 10, 15, 20,....

8.What is non continuous arithmetic progression?

The non consistent arithmetic development in identified as a sequence having typical differences other than 0. For example, 1, 2, 3, 4 etc.

9.How do you find the nth hatchet of a succession with different differences?

The measures to discover the nth ax of a sequence v different differences are:

We take the difference between the consecutive terms.If the difference among consecutive termsis not constant, we examine the change in differenceoccurring.If the adjust in difference emerging is a, climate the nth term is offered as (\(\dfraca2\))n2.

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

An arithmetic succession is identified as a sequence wherein the usual difference among the succeeding terms is constant.An arithmetic series the amount of all terms of an arithmetic sequence.