Question

If every term of an A. P. is multiplied by the same quantity, then the
resulting sequence is an......​

Answer 1

Answer:

A.P.

Step-by-step explanation:

=(a, a+d, a+2d.......)+(a, a+d, a+2d........)

=2a,2(a+d),2(a+2d)............

Hence It will become an A. P.

Answer 2

Answer :

AP(Arithmetic Progression)

Note :

★ A.P. (Arithmetic Progression) : A sequence in which the difference between the consecutive terms are equal is said to be in A.P.

★ If a1 , a2 , a3 , . . . , an are in AP , then

a2 - a1 = a3 - a2 = a4 - a3 = . . .

★ The common difference of an AP is given by ; d = a(n) - a(n-1) .

★ The nth term of an AP is given by ;

a(n) = a1 + (n - 1)d .

★ If a , b , c are in AP , then 2b = a + c .

★ The sum of nth terms of an AP is given by ; S(n) = (n/2)×[ 2a + (n - 1)d ] .

★ The nth term of an AP can be also given by ; a(n) = S(n) - S(n-1) .

★ A linear polynomial in variable n always represents the nth term of an AP .

★ A quadratic polynomial in variable n always represents the sum of n terms of an AP .

★ If each terms of an AP is multiplied or divided by same quantity , then the resulting sequence is an AP .

★ If same quantity is added or subtracted in each term of an AP then the resulting sequence is an AP .

Solution :

Let a1 , a2 , a3 are in AP such that ;

a1 = a

a2 = a + d

a3 = a + 2d

where d is the common difference .

Now ,

Multiplying each terms of the AP be the same quantity (say k) , we have ;

a1' = ka

a2' = k(a + d)

a3' = k(a + 2d)

Now ,

a2' - a1' = k(a + d) - ka

= k(a + d - a)

= kd

a3' - a2' = k(a + 2d) - k(a + d)

= k(a + 2d - a - d)

= kd

Clearly ,

The difference between consecutive terms of the new sequence is same ( with the new common difference d' = kd ) .

Hence ,

The resulting sequence is an AP .