Let 'k' is added in each term of the above sequence we get,

$a_{1} + k,a_{2} + k ,a_{3} + k ,a_{4} + k ,...,a_{n}$ + k -------(i)

$a_{n} = a_{1},a_{2},a_{3},a_{4},...,a_{n}$

(i) ⇒ $a_{n}$ + k where n = 1,2,3,4,...

Let us consider the above equation as $b_{n}$

$b_{n} = a_{n}$ + k

So $b_{n}$ is a new sequence which can be represented as

$b_{1},b_{2},b_{3},b_{4},...,b_{n}$

$b_{n + 1} - b_{n} = [a_{n + 1} + k ] -[a_{n} + k] = (a_{n + 1} - a_{n}$) = d

So common difference of new sequence is also constant so even if the constant is added to each term of an A.P. then the resulting sequence is also an A.P.

If we add 3 to each term of A.P. , we will get

(2 +3), (4 + 3 ), (6 + 3), (8 + 3),...

5,7,9,11,.... is also an A.P sequence.

2,4,6,8,... is an A.P

If we subtract 3 to each term of A.P. , we will get

(2 - 3), (4 - 3 ), (6 - 3), (8 - 3),...

-1,1,3,5,.... is also an A.P sequence.

Example : 1,3,5,7,... is an A.P

If we multiply each term by 2, we will get

(1 x 2), (3 x 2 ), (5 x 2), (7 x 2),...

2,6,10,14,.... is also an A.P sequence.

Example : 12,14,16,18,... is an A.P

If we divide each term by 2, we will get

(12 $\div $ 2), (14 $\div $ 2 ), (16 $\div $2), (18 $\div $ 2),...

6,7,8,9,.... is also an A.P sequence.

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