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Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. `3, 3 + sqrt2, 3 + 2sqrt2, 3 + 3sqrt2`

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#### Solution

`3, 3 + sqrt2, 3 + 2sqrt2, 3 + 3sqrt2`

Here,

a_{2} - a_{1} = `3 + sqrt2 - 3 = sqrt2`

a_{3} - a_{2} = `(3 + 2sqrt2) - (3 + sqrt2) = sqrt2`

a_{4} - a_{3} = `(3 + 3sqrt2) - (3 + 2sqrt2) = sqrt2`

⇒ a_{n+1} - a_{n} is same every time.

Therefore, `d = sqrt2` and the given numbers are in A.P.

Three more terms are

a_{5} = `(3 + sqrt2) + sqrt2 = 3 + 4sqrt2`

a_{6} = `(3 + 4sqrt2) + sqrt2 = 3 + 5sqrt2`

a_{7} = `(3 + 5sqrt2) + sqrt2 = 3 + 6sqrt2`

Concept: Arithmetic Progression

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