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

p, p + 90, p + 180 p + 270, ... where p = (999)^{999}

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#### Solution

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*)

p, p + 90, p + 180 p + 270, ... where p = (999)^{999}

Here

First term (*a*) = *p*

`a_1 = p + 90`

`a_2 = p + 180`

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

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

Here

`a_1 - a= p + 90 - p`

= 90

Also

`a_2 - a_1 = p + 180 - p - 90`

= 90

Since `a_1 - a = a_2 - a_1`

Hence, the given sequence is an A.P and its common difference is d = 90

Concept: Arithmetic Progression

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