# 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, nth term , sum to first nth term and Properties of Arithmetic Progression.

## Arithmetic Progression Definition, Properties, Formulas  | Allmathtricks

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)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 nth  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 nth term

rn = r + (n -1) d

rn 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 nth terms of an Arithmetic progression

Tn = a + ( n-1) d

Here Tn = nth 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 nth  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 nth term of this A.P is an = 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

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

Sn = (n/2) ( a + an)

### 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,, . . . . . an-1, an are in A.P then an, an-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 an > an-1  ( Here an is a nth  term of above A.P)

If d < 0, then an < an-1  ( Here an is a nth  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 4.  The nth 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  an = Sn – Sn-1

5. If r1, r2, r3, r4, . . . . .  rn 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=  r2 + rn-1   = r3 + rn-2   = r4 + rn-3   = r5 + rn-4

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

i) rn, rn-1, rn-3,  . . . . .  r2, rare in A.P

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

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

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

i) a1+r1, a2+r2, a3+r3, . . . . .  an+rn  is in A.P

ii) a1r1, a2r2, a3r3,  . . . . .  anrn  is not in A.P

iii) a1/r1, a2/r2, a3/r3,  . . . . .  an/rn  is not in A.P Related Articles

Sequence and series

Arithmetic Progression Formulas

Arithmetic Progression Examples

Arithmetic Mean formula with Examples

Geometric Progression formulas

Geometric Progression Examples

Harmonic Progression

Types of Angle Pairs

Factorizing Algebraic Expressions

Logarithm Applications