Arithmetic Progression

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.

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

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