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AP IN MATHS | DEFINITION, nth Term ! CLASS 10TH #CBSE #BOARDEXAM


An arithmetic progression (AP) is a sequence of numbers where the differences between every two consecutive terms are the same. In this progression, each term, except the first term, is obtained by adding a fixed number to its previous term. This fixed number is known as the common difference and is denoted by 'd'. The first term of an arithmetic progression is usually denoted by 'a' or 'a1'.

For example, 1, 5, 9, 13, 17, 21, 25, 29, 33, ... is an arithmetic progression as the differences between every two consecutive terms are the same (as 4). i.e., 5 - 1 = 9 - 5 = 13 - 9 = 17 - 13 = 21 - 17 = 25 - 21 = 29 - 25 = 33 - 29 = ... = 4. We can also notice that every term (except the first term) of this AP is obtained by adding 4 to its previous term. In this arithmetic progression:

a = 1 (the first term)
d = 4 (the "common difference" between terms)
Thus, an arithmetic progression, in general, can be written as: {a, a + d, a + 2d, a + 3d, ... }.

Arithmetic Progression

In the above example we have: {a, a + d, a + 2d, a + 3d, ... } = {1, 1 + 4, 1 + 2 × 4, 1 + 3 × 4, ... } = {1, 5, 9, 13, ... }

Arithmetic Progression Formula (AP Formulas)
For the first term 'a' of an AP and common difference 'd', given below is a list of arithmetic progression formulas that are commonly used to solve various problems related to AP:

Common difference of an AP: d = a2 - a1 = a3 - a2 = a4 - a3 = ... = an - an-1
nth term of an AP: an = a + (n - 1)d
Sum of n terms of an AP: Sn = n/2 (2a + (n - 1) d) = n/2 (a + l), where l is the last term of the arithmetic progression.
The following image comprehends all AP formulas.

Arithmetic Progression Formula or A P formulas

Common Terms Used in Arithmetic Progression
From now on, we will abbreviate arithmetic progression as AP. An AP generally is shown as follows: a1, a2, a3, . . . It involves the following terminology.

First Term of Arithmetic Progression:
As the name suggests, the first term of an AP is the first number of the progression. It is usually represented by a1 (or) a. For example, in the sequence 6, 13, 20, 27, 34, . . . . the first term is 6. i.e., a1 = 6 (or) a = 6.

Common Difference of Arithmetic Progression:



Therefore, the 102nd term of the given AP 6, 13, 20, 27, 34, .... is 713. Thus, the general term (or) nth term of an AP is referred to as the arithmetic sequence explicit formula and can be used to find any term of the AP without finding its previous term.

The following table shows some AP examples and the first term, the common difference, and the general term in each case.

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Nth Term of AP – Definition, Formula, Examples, FAQs
Home » Math Vocabulary » Nth Term of AP – Definition, Formula, Examples, FAQs
What Is the nth Term of an AP?
The nth Term of an AP Formula
How to Find the nth Term of an AP
Solved Examples on the nth Term of an AP
Practice Problems on the nth Term of an AP

Tn= a+(n – 1)d
where:

Tn is the nth term of the AP

a is the first term of the AP

n indicates the position of the term

d is the common difference between consecutive terms of the AP.
Derivation of the nth Term of an AP
An arithmetic progression (A.P) is a sequence of numbers where each term is obtained by adding a fixed amount, known as the common difference, to the preceding term.
This formula represents the value of the nth term in AP whose first term is a and the common difference is d.
How to Find the nth Term of an AP*
Let’s understand the steps to find the nth term of an Arithmetic Progression (AP) with the help of an example.

Formula: nth term Tn=a+(n-1)d
Facts about the nth Term of an AP Formula
The common difference (d) remains constant throughout the entire arithmetic progression. It’s the fixed amount that each term increases or decreases by.
The first term (a1) of the AP is the value at the first position, i.e., a1=a1+(1-1)d, which simplifies to a1.
The nth term represents the value of the term at the nth position in the sequence. It’s the term that’s n steps away from the first term. cbse board exam
The formula works for negative terms as well. If the common difference is negative, adding it will cause the terms to decrease as you move along the sequence.
The formula can also be used to find terms with non-integer values. As long as the common difference and initial term are real numbers, the formula remains valid.
Conclusion
In this article, we learned about finding the nth term of an arithmetic progression (AP). ://www.instagram.com/perfect_maths_solution_73?igsh=dzZjbHFpM2FhMGFz
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