CBSE Class 10 Arithmetic Progression Important Formulas and concepts for exams

CBSE Class 10 Arithmetic Progression Important Formulas and concepts for exams. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

SEQUENCE

An arrangement of numbers in a definite order according to some rule is called a sequence. In other words, a pattern of numbers in which succeeding terms are obtained from the preceding term by adding/subtracting a fixed number or by multiplying with/dividing by a fixed number, is called sequence or list of numbers.

e.g. 1,2,3,4,5

A sequence is said to be finite or infinite accordingly it has finite or infinite number of terms. The various numbers occurring in a sequence are called its terms. 

ARITHMETIC PROGRESSION ( AP ).

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

This fixed number is called the common difference of the AP. It can be positive, negative or zero. Let us denote the first term of an AP by a1, second term by a₂, . . ., nth term by an and the common difference by d. Then the AP becomes a₁, a₂, a₃, . . ., an.

So, a₂ – a₁ = a₃ – a₂ = . . . = a_n – a_n –1 = d.

The general form of an arithmetic progression is given by a, a + d, a + 2d, a + 3d, . . .

where a is the first term and d the common difference.

nth Term of an AP

Let a₁, a₂, a₃, . . . be an AP whose first term a1 is a and the common difference is d.

Then,

the second term a₂ = a + d = a + (2 – 1) d

the third term a₃ = a₂ + d = (a + d) + d = a + 2d = a + (3 – 1) d

the fourth term a₄ = a₃ + d = (a + 2d) + d = a + 3d = a + (4 – 1) d

. . . . . . . .

. . . . . . . .

Looking at the pattern, we can say that the nth term a_n = a + (n – 1) d.

So, the nth term an of the AP with first term a and common difference d is given by

                                      a_n = a + (n – 1) d.

an is also called the general term of the AP. If there are m terms in the AP, then am represents the last term which is sometimes also denoted by l. 

nth Term from the end of an AP

Let the last term of an AP be ‘l’ and the common difference of an AP is ‘d’ then the nth term from the end of an AP is given by 

                                l_n = l – (n – 1) d.

Sum of First n Terms of an AP

 

The sum of the first n terms of an AP is given by

 

where a = first term, d = common difference and n = number of terms.

Also, it can be written as

where an = nth terms

or

where l = last term

This form of the result is useful when the first and the last terms of an AP are given and the common difference is not given..

 

Problems based on finding an if Sn is given.

 

Find the nth term of the AP, follow the steps:

 

◊ Consider the given sum of first n terms as Sn.

 

◊ Find the value of S1 and S2 by substituting the value of n as 1 and 2.

 

◊ The value of S1 is a1 i.e. a = first term and S2 – S1 = a2

 

◊ Find the value of a2 – a1 = d, common difference.

 

◊ By using the value of a and d, Write AP. Problems based on finding Sn if an is given. 

 

Find the sum of n term of an AP, follow the steps:

 

◊ Consider the nth term of an AP as an.

 

◊ Find the value of a1 and a2 by substituting the value of n as 1 and 2.

 

◊ The value of a1 is a = first term.

 

◊ Find the value of a2 – a1 = d, common difference.

 

◊ By using the value of a and d, Write AP.

 

◊ By using Sn formula, simplify the expression after substituting the value of a and d.

 

Arithmetic Mean

 

If a, b and c are in AP, then ‘b’ is known as arithmetic mean between ‘a’ and ‘c’ b=a+c/2 i.e AM between 'a' and 'c' is a+c/2

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