Arithmetic sequence properties

We will discuss arithmetic sequence properties which will be used in solving different types of problems.
Properties of an A.P
Property I : If a constant is added to each term of an A.P; the resulting sequence is also an A.P.
Proof : Let $a_{1},a_{2},a_{3},a_{4},...,a_{n}$ is arithmetic sequence with common difference 'd'.
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.

Example : 2,4,6,8,... is 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.

Property II : If a constant is subtracted from each term of an A.P; the resulting sequence is also an A.P.
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.

Property III : If each term of an A.P is multiplied by a constant, then the resulting sequence is also an A.P.
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.

Property IV : If each term of an A.P is divided by a constant, then the resulting sequence is also an A.P.
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.

Property IV : If the nth term is in linear form i.e. an = an + b then the sequence is in A.P.

Property VI : If the sum of the first term of the sequence is of the for a$n^{2}$ + bn where 'a' and 'b' are any constant then the given sequence is in A.P.

Property VII : If the terms are selected at a regular interval the the given sequence is in A.P.

Property VIII : If three consecutive numbers a,b and c are in A.P. then the sum of two numbers is twice the sum of the middle number i.e.2b = a + c

11th grade math

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