In mathematics, an **Arithmetic Progression**(AP) or **Arithmetic Sequence** is a sequence of numbers such that the difference between the consecutive terms is constant and is known as **common difference**. For example, the sequence 2, 4, 6, 8, … is an arithmetic sequence with the common difference 2.

We can find the common difference of an AP by finding the difference between any two adjacent terms.

Index

**History**

Evidence shows that **Babylonians**, some **4000 years ago**, knew arithmetic and geometric sequences. According to *Boethius*, arithmetic and geometric sequences were known to early Greek writers.

Among the Indian mathematicians, **Aryabhatta** was the first to give the formula for the sum of squares and cubes of natural numbers in his famous work ‘**Aryabhatiyam**’.

He also gave the formula for finding the sum to ‘n’ terms of an arithmetic sequence starting with the \(p^{th}\) term.

**Noted: **Indian mathematicians **Brahmgupta**, **Mahavira**, and **Bhaskara** are also considered to give the sum of squares and cubes.

**Terms in Arithmetic Progression**

A sequence \(a, a_1, a_2, a_3, …, a_n\) is called an arithmetic sequence or arithmetic progression.

Where,

\(a\) is the first term.

\(a_1\) is the second term, and can be given by \(a_1 = a + d\),

\(a_2\) is the third term, with \(a_2 = a_1 + d = a + 2d\)

\(d\) is the common difference between two consecutive terms and it remains the same throughout a particular series.

We can write this arithmetic sequence as \(a, a + d, a + 2d, a + 3d, …, a + (n-1)d\)

Here we can get the **\(n^{th}\)** term of the** arithmetic sequence** as

\( a_n = a + (n-1)d\)

Where,**\( a_n\)** is the \(n^{th}\) term**n** is the number of terms**a** is the first term**d** is the common difference

NOTE: If a constant is **added**, **subtracted**, **multiplied **(or) **divided **(by a **non-zero constant**) to each term of an A.P., the resulting sequence is also an A.P.

To find if three terms are in A.P.

**a + c = 2b**, here **a, b & c **are** 1st, 2nd **and** 3rd** term respectively.

If this is correct then we can say that those three terms are in A.P.

**Example:** In sequence 1, 4, 7

1 + 7 = 8, i.e 2(4)

Here we can say that the terms 1, 4, 7 are in A.P.

**Sum of Arithmetic Progression**

The sum of the members of a finite arithmetic progression is called an **arithmetic series**.

**Derivation**

\(S_n = a + (a + d) + (a + 2d) + … + (a + (n-2)d) + (a + (n-1)d) …(i)\\

S_n = (a + (n-1)d) + (a + (n-2)d) + … + (a + 2d) + (a + d) + a …(ii)\\

\mbox{adding (i) & (ii)}\\

\mbox{All terms including ‘d’ cancels out leaving}\\

2S_n = n(a + a_n)\\

S_n = \frac{n}{2}(a + a_n)\\

\mbox{because } a_n = a + (n-1)d\\

S_n = \frac{n}{2}(2a + (n-1)d)\\

\)

**Arithmetic Progression Applications**

**Sequences** are useful in several mathematical disciplines for studying **functions**, **spaces**, and other mathematical structures using the **convergence properties of sequences**. In particular, sequences are the basis for series, which are important in **differential equations** and **analysis**.

**Questions**

Let us look at some **arithmetic progression questions**:

**Question 1.** If the sum of *n* terms of an A.P. is \((pn + qn^2)\), where *p* and *q* are constants, find the common difference.

**Solution.**

As we know the formula,

\(S_n = \frac{n}{2}(2a + (n-1)d)\\

\mbox{From the question we can say that}\\

\frac{n}{2}(2a + (n-1)d) =pn +qn^2\\

\frac{n}{2}(a + nd-d ) = pn + qn^2\\

na + n^2\frac{d}{2} – n \frac{d}{2} = pn + qn^2\\

n^2(\frac{d}{2} – q) + n(a-p-\frac{d}{2}) = 0\\

\)

Comparing the coefficients of \(n^2\)

We can say that \(\frac{d}{2} = q\) (OR) d = 2q

Therefore, difference in A.P. will be 2q.

**Question 2.** In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that \(20^{th}\) term is –112.

**Solution.**

First term, a = 2

Sum of first 5 terms, S_5 = -112

Therefore, \(10 + 10d = \frac{1}{4}(10 + 35d)\)

=> 40 + 40d = 10 + 35d

=> d = -6

Therefore, \(a_{20} = 2 + (20 – 1)(-6) = (-112)\)

Therefore, the 20th term is -112.

More On Progression: Harmonic Progressions

**FAQs**

**What is the formula to calculate ‘D’?**The formula to calculate

\(D = a_n – a_{n-1}\)

**What is the arithmetic mean between 4 & 24?**The arithmetic mean is given by \(\frac{a+b}{2}\)

=> A.M. of 4 & 24 is 14.

**Difference between arithmetic progression and arithmetic series?**Arithmetic Progression is a series with a common difference up to ‘\(n^{th}\)’ term. The arithmetic Series is the sum of the elements of Arithmetic Progression.

**What is the formula for the sum of the square of no.s till n terms?**\(1^2 + 2^2 + 3^2 + 4^2 +… + n^2 = \frac{n(n+1)(2n+1)}{6}\)

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