Question

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

10, 10 + 2⁵, 10 + 2⁶, 10 + 2⁷,...

Answer 1

In the given problem, we are given various sequences.

We need to find out that the given sequences are an A.P or not and then find its common difference (d)

10, 10 + 2⁵, 10 + 2⁶, 10 + 2⁷,...

Here

First term (a) = 10

`a_1 = 10 + 2^5`

`a_2 = 10 + 2^6`

`a_3 = 10 + 2^7`

Now, for the given to sequence to be an A.P,

Common difference (d) = `a_1 - a = a_2 - a_1`

Here

`a_2 - a_1 = 10 + 2^6 - 10 - 2^5`

= 64 - 32

= 32

Also

`a_3 - a_2 = 10 + 2^7 - 10 - 2^6`

= 128 - 64

= 64

Since `a_1 - a != a_2 - a_1`

Hence, the given sequence is not an A.P