Arithmetic Progression Formula for Class 10 and Competitive Exams

In this session explained about Basic concepts of Sequence and series, introduction and formulas for arithmetic progression like Common Difference, Finite and infinite arithmetic Progressions, n^th term , sum to first n^th term and Properties of Arithmetic Progression.

Arithmetic Progression Definition, Properties, Formulas | Allmathtricks

Contents

1 Arithmetic Progression Definition, Properties, Formulas | Allmathtricks
- 1.1 Arithmetic Progression
  - 1.1.1 Common Difference
  - 1.1.2 Formula for finding the sum to first nth term an A.P
- 1.2 Properties of Arithmetic Progression

Sequence following specific patterns are called progressions.

Basic concepts of sequence and series

Progressions are of three types. They are

1. Arithmetic Progression

2. Geometric Progression

3. Harmonic Progression

Arithmetic Progression

Arithmetic Progression can be defined as, a sequence of numbers is obtained by adding a fixed number “d” to the preceding term except the first term.

i.e A list of numbers X₁, X₂, X₃, X₄ . . . . . . . is an A.P, If the differences X₂-X₁, X₃-X₂, X₄-X₃, . . . . given the same value

Each of the number in the list of an arithmetic progression is called a term of that A.P.

The general form of an A.P is a, a+d, a+2d, a+3d . . . . . . ( Here “d” is the common difference and “a” is the first term )

Common Difference

The difference between two succeeding terms of an Arithmetic Progression is called common difference. The difference value can be positive,negative or zero.

Examples:

1) Rainfall of the last week ( in mm) 12, 10, 08, 06, 04, 02, 0

2) Temperature record for the last week ( in Deg.C) 30, 31, 32, 33, 34, 35, 36

3) 2, 4, 6, 8, 10, . . . . . . . . . . . .

4) 0, -5, -10, -15, -20, . . . . . . . . . . .

Finite and infinite arithmetic Progressions

In the above examples no. 1 & 2 having finite number of terms. Such arithmetic Progressions is called finite A.P. In the same way above examples no. 3 & 4 having infinite number of terms. So they are called infinite arithmetic Progressions

Formula for finding the n^th term or general term of AP formula

Let r₁, r₂, r₃, r₄, . . . . . be an AP, Here first term ‘r₁‘ is “r” and common difference is “d”

Then,

First term r₁ = r = r + (1 -1 ) d

Second term r₂ = r + d = r + ( 2-1)d

Third term r₃ = r₂ + d = r + ( 2-1)d +d = r + (3 -1) d

Fourth term r₄ = r₃ + d = r + (3 -1) d +d = r + (4 -1) d

Firth term r₅= r₄ + d = r + (4 -1) d +d = r + (5 -1) d

Similarly use same pattern for n^th term

r_n = r + (n -1) d

r_n is also called general term of the A.P

Using the above formula we can find different terms of the A.P

General notation of n^th terms of an Arithmetic progression

Tn = a + ( n-1) d

Here Tn = n^th of an A.P ; a = first term of the A.P ; d = common difference

If number of terms in an A.P is “ r” and last term is “L” then,

L = a + ( r – 1)d

Formula for finding the sum to first n^th term an A.P

Let a, a+d, a+2d, a+3d, . . . . . . . . . be an AP, Here first term “a” and common difference is “d‘

The n^th term of this A.P is a_n = a + ( n-1) d

Let Sum of “n” terms of an A.P is ” Sn“

Sn = a + (a+d) + (a+2d) + ( a+3d) + . . . . . . . . . + [ a + ( n-1) d ]

Now write Sn from last term of A.P

Sn = [ a + ( n-1) d ] + [ a + ( n-2) d ] + [ a + ( n-3) d ] + . . . . . . . . . . . + a

Adding Sn +Sn

⇒ 2Sn = [ 2a + ( n-1) d ] + [ 2a + ( n-1) d ] + [ 2a + ( n-1) d ] +. . . . . . . . . . .+ [ 2a + ( n-1) d ] ( n times)

⇒ 2Sn = n [ 2a + ( n-1) d ]

⇒ $Sn \ = \frac{n}{2} \ [2a \ + \ (n-1)d)]$

⇒ Sn = (n/2) [ a + a + ( n-1) d ] = (n/2) ( first term + n^th term)

⇒ Sn = (n/2) ( a + a_n)

Properties of Arithmetic Progression

1. Reversed Sequence

If the Arithmetic Progression a, a+d, a+2d, a+3d . . . . . . m is reversed to m, m-d , m-2d, m-3d, . . . . a, then the common difference changes to negative of original common difference, and the reversed sequence is also A.P.

i.e a₁, a₂, a₃,, . . . . . a_n-1, a_nare in A.P then a_n, a_n-1, . . . . . a₃, a₂, a₁, are in A.P

Example:

1, 3, 5, 7, 9 is an A.P with common difference ” 2″

Now 9, 7, 5, 3, 1 is also an A.P with common difference “-2 “

2. If the Arithmetic Progression a, a+d, a+2d, a+3d . . . . . . , then

If d >0, then a_n > a_n-1 ( Here a_n is a n^th term of above A.P)

If d < 0, then a_n < a_n-1 ( Here a_n is a n^th term of above A.P)

3. If the sum of the first two terms is “a” and the sum of last two terms is “b” , number of terms is “n”

then the sum of the A.P is $n \ \left [ \ \frac{ a + b}{4} \right ]$

4. The n^th term of Arithmetic Progression is the difference of the sum to first “n” terms and sum of first (n-1) terms of it.

i.e a_n = Sn – Sn-1

5. If r₁, r₂, r₃, r₄, . . . . . r_n be an finite A.P, then the sum of the terms equidistant from the beginning and the end is always same and is equal to the sum of the first and last term.

i.e r₁ + r_n= r₂ + r_n-1= r₃ + r_n-2= r₄ + r_n-3= r₅ + r_n-4

6. If r₁, r₂, r₃, r₄, . . . . . r_n be an finite A.P, then

i) r_n, r_n-1, r_n-3, . . . . . r_2,r₁are in A.P

ii) r₁ ± k, r₂ ± k, r₃± k, r₄± k, . . . . . r_n± k are in A.P ( Where k ∈ R )

iii) kr₁, kr₂, kr₃, kr₄, . . . . . kr_n are in A.P ( Where k ∈ R – {0} )

7. If r₁, r₂, r₃, r₄, . . . . . r_n & a₁, a₂, a₃, a₄, . . . . . a_n are two finite A.P, then