Question

If each term of an arithmetic sequence is multiplied by a fixed​ number, will the resulting sequence always be an arithmetic​ sequence?

Answer 1

Answer:

yes correct, also when you are adding or subtracting

Step-by-step explanation:

Answer 2

Answer:

Yes, the resulting sequence will always be an arithmetic sequence.

Step-by-step explanation:

We know that any arithmetic sequence is a sequence in which each term of the sequence is increased by a fixed constant as compared to the preceding term of the sequence.

This fixed constant is known as common difference,

i.e.  $d=a_n-a_{n-1}$

where

$a_n$ is the nth term of the sequence.

Now, when  each term of an arithmetic sequence is multiplied by a fixed​ number let a'.

Let $a_n'$ is the nth term of the new sequence.

Then we have:

$a_1'=a'a_1$

$a_2'=a'a_2$

$a_3'=a'a_3$

and so on

Also,

$a_{n}'-a_{n-1}'=a'a_n-a'a_{n-1}\\\\a_{n}'-a_{n-1}'=a'(a_n-a_{n-1})\\\\a_{n}'-a_{n-1}'=a'd$

for each n.

Hence, the sequence will again be an arithmetic sequence.

          ( Since, the sequence again vary by a fixed constant )