**Common difference in Arithmetic Progression**

Common difference in arithmetic progression is denoted by **d .**It is the difference between the successive term and its preceding term. It is always constant or same for arithmetic progression. In short we can say that, if in a given sequence if the common difference is constant then we can say that the given sequence is in A.P.

If the sequence is $a_{1}, a_{2}, a_{3}, a_{4}$,...

So, $a_{2} - a_{1}$ = d

$a_{3} - a_{2}$ = d

$a_{4} - a_{3}$ = d and so on.

**Example :**7,11,15,19,23,...

Here, 11 - 7 = 4

15 -11 = 4

19 - 15 = 4

23 - 19 = 4

From the above we can see that the difference between the successive terms is same (constant) which is 4, so we can say that the given sequence is in A.P.

If the 1st term and the common difference 'd' is given then we can make an arithmetic sequence.

## Examples on Common difference in Arithmetic Progression

1) Write the sequence if the first term is 8 and the common difference is -3.**Solution :**Here,

**$a_{1}$ = 8**

d = -3

As we know that,

$a_{2} - a_{1}$ = d

$a_{2} = a_{1}$ + d

∴ $a_{2}$ = 8 + (-3)

**$a_{2}$ = 5**

Now, $a_{3} - a_{2}$ = d

∴ $a_{3} = a_{2}$ + d

$a_{3}$ = 5 + (-3)

**$a_{3}$ = 2**

$a_{4} - a_{3}$ = d

∴ $a_{4} = a_{3}$ + d

$a_{4}$ = 2 + (-3)

**$a_{4}$ = -1**

∴ Sequence is 8, 5, 2, -1,...

2) Write an A.P. whose first term is 10 and the common difference is 3 .

**Solution :**We know that if 'a' is the first term and d is the common difference , then the arithmetic progression is

a , a + d , a + 2d, a + 3d, a + 4d, ...

Here, a = 10 and d = 3

10, (10 + 3), (10 + 2 x 3), (10 + 3 x 3), ( 10 + 4 x 3),...

10, 13, ( 10 + 6), (10 + 9), ( 10 + 12), ...

So, the arithmetic progression is

10, 13, 16, 19, 22,...

3) Write an A.P. whose first term and common difference are -1.25 and - 0.25 respectively.

**Solution :**We know that if 'a' is the first term and d is the common difference , then the arithmetic progression is

a , a + d , a + 2d, a + 3d, a + 4d, ...

Here, a = -1.25 and d = -0.25

-1.25 , (-1.25 + -0.25 ), ( -1.25 + 2 x -0.25 ), ( -1.25 + 3 x -0.25 ), ( -1.25 + 4 x -0.25 ),...

-1.25, -1.50, ( -1.25 + - 0.5 ), (-1.25 + - 0.75), (-1.25 + -1), ...

-1.25, -1.75, - 2, -2.25,... So, the arithmetic progression is

-1.25, -1.75, - 2, -2.25,...

**Practice Questions**

I. Find the common difference from the given sequence.

a) 1, -3, -7, -11, -15,...

b) 8,4,0,-4,-8,...

c) -5,-3,-1,1, 3,...

II. Check the given sequence is in A.P or not. a) 3, 6, 12, 24, ...

b) $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}$, ...

c) 1, 1.7, 2.4, 3.1, ...

d) $1^{2}, 3^{2}, 5^{2}, 7^{2}$, ...

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