Arithmetic Progressions Class 10 Notes CBSE Maths Chapter 5 (Free PDF Download)

Definition of Arithmetic Progression:

• An arithmetic progression is a sequence of numbers, obtained by adding a fixed number to the preceding term starting from the first term such that the difference between each consecutive term remains the same.

• Each of the numbers in the list is called a term and the fixed number is called the common difference of the AP which can be any integer.

• For example: $2,5,8,11....$ having a common difference of $3$.

General term of an AP:

1. The general form of an AP is: 

$a\text{ },a+d\text{ },a+2d\text{ },a+3d\text{ },....,a+(n-1)d$

2. An AP with a finite number of terms is called a finite AP having           $a+(n-1)d$ as the last term. 

For example:

Finite AP: $1,3,5,7,....,25$

An AP which neither has a finite number of terms nor has a last term is called an infinite AP.

For example: 

Infinite AP: $2,4,6,8.....\infty $

3. The ${{n}^{th}}$ term of the AP: $an=a+(n-1)d$, where $a$ is the first term of the sequence and $d$ is the common difference.

The Second term: ${{a}_{2}}=a+(2-1)d=a+d$

Similarly, the third term ${{a}_{3}}=a+(3-1)d=a+2d$

The fourth term ${{a}_{4}}=a+(4-1)d=a+3d$ and so on till the last term.

Example 1: 

An AP has a first term $3$, common difference $4$. Find the third and fifth term of the AP.

Ans: 

$a=3,\text{ d}=\text{4}$ 

${{\text{a}}_{3}}=3+(3-1)4$

${{a}_{3}}=11$

Similarly, 

${{a}_{5}}=3+(5-1)4$

${{a}_{5}}=19$

4. ${{n}^{th}}$ term of an AP from the end:  $tn=L-(n-1)d$, where $\text{L}$ is the last term of the AP.

Example 2:

An AP has a common difference $2$ and last term $24$. Find the fourth term of the AP from the end.

Ans:

$d=2,\text{ L}=2\text{4}$ 

${{t}_{4}}=24-(4-1)2$

${{t}_{4}}=18$

Sum of the terms of an AP:

1. Sum of $n$ terms of an AP if first term and common difference is given:

$S=\dfrac{n}{2}(2a+(n-1)d)$

2. Sum of $n$ terms of an AP if first term and last term $l$ is given:

$S=\dfrac{n}{2}(a+l)$

Example 3:

Find the sum of first $10$ terms of the AP $1,4,7,10.....34.$

Ans:

$S=\dfrac{10}{2}(2\times 1+(10-1)3)$

$=5(2+27)$

$=5\times 29$

$=145$

Arithmetic Progression

An arithmetic progression is a sequence of numbers that differ from each other by a common difference. For example, the sequence 3, 6, 9, 12, ..... is an A.P. with a common difference of 3.

Common Difference

The difference between the two consecutive terms of an A.P. is known as the common difference. For example, in the sequence 3, 6, 9, 12...., the common difference is 3.

The classification of the common difference:

• Positive, when the A.P. is increasing.

• Zero, when the A.P. is constant.

• Negative, when the A.P. is decreasing.

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General Form of an Arithmetic Progression

Say the terms a1, a2, a3……an are in A.P. If the first term is ‘a’ and its common difference is ‘d’. Then, the terms can also be expressed as follows.

1st term a1 = a

2nd term a2 = a + d

3rd term a3 = a + 2d

Therefore, we can also represent arithmetic progressions as:

a, a + d, a + 2d, ……

This representation is called the general form of an Arithmetic Progression.

Finite and Infinite A.P.

• In the finite A.P., the numbers of terms are finite, and the last term of the A.P. exists.

• In the infinite A.P., the number of terms is infinite, and the last term of an A.P. doesn’t exist.

Sum of Arithmetic Progressions

The sum of n terms of an A.P. with ‘a’ as its first term and ‘d’ as its common difference is given by:

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

Arithmetic Mean

Arithmetic Mean is simply the average of two numbers. If we have two numbers n and m, we can add a number L in between them so that the three numbers form an arithmetic sequence like n, L, m. In this case, the number L is the arithmetic mean of the numbers n and m. On the basis of the properties of Arithmetic Progression, we may say:

L – n = m – L, that is, the arithmetic mean of n and m.

\[L = \frac{n + m}{2}\]

Properties of Arithmetic Progressions

• If the same number is added or subtracted from each A.P. term, the resulting terms in the sequence are also in A.P. with the same common difference.

• If each term in A.P. is divided or multiplied by the same non-zero number, the resulting series is also in A.P.

• Three numbers x, y, and z will be an A.P. if 2y = x + z.

• A series is an A.P. if the nth term is a linear expression.

• If we pick terms from the A.P. in the regular interval, these selected terms will also make an A.P.

• If the terms of an arithmetic progression are increased or decreased with the same amount, the resulting sequence will also be an arithmetic progression.

Solved Examples

1. Find the missing term in the following AP. 5, x, 13.

Ans: Three numbers x, y, z are in AP if 2y = x + z. Given x = 5, z = 13 therefore y = (x+z)/2.

By substituting values we get:

y = (5 + 13)/2 = 18/2 = 9

Therefore, the missing term is 9.

2. Write the first five terms of A.P., when a = 23 and d = - 5.

Ans: The general form of A.P. is a, a+d, a+2d …

When a = 23 and d = -5, the first five terms of AP are 23, 23+(-5), 23 + 2(-5), 23 +3(-5), 23 + 4(-5).

Hence, the first five terms of AP are 23, 18, 13, 8 and 3.

Important Points on Arithmetic Progressions

• If each term of the A.P. is increased, decreased, multiplied, or divided by the same non-zero constant, the resulting sequence would also be in A.P.

• In the A.P., the number of terms equidistant from start to end will be constant.

• In order to solve most of the problems related to A.P., the terms can be conveniently taken as:

3 Terms: (a - d), a, (a + d).

4 Terms: (a - 3d), (a - d), (a + d), (a + 3d).

Important Formulas

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Conclusion

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