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If the following series form an $AP$ , find the common difference $d$ and the next $3$ terms. $3, 3 +$ $\sqrt{2}$ , $3 +$ 2 $\sqrt{2}$ , $3 +$ 3 $\sqrt{2}$ , …

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It is not an

AP

d = \sqrt{2}

t_{5} = 3 + 4 \sqrt{2}

t_{6} = 3 + 5 \sqrt{2}

t_{7} = 3 + 6 \sqrt{2}

d = \sqrt{3}

t_{5} = 3 + 4 \sqrt{2}

t_{6} = 4 + 5 \sqrt{2}

t_{7}

= 5 +

6 \sqrt{2}

None of these

answer is B.

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Detailed Solution

The given series is,

3, 3 +

\sqrt{2}

3 +

\sqrt{2}

3 +

\sqrt{2}

, …
It is clear that the given sequence is an AP having first term,

a = 3

and common difference,

d = \sqrt{2}

.
Thus,

n^{th} term = a_{n} = a + (n - 1) d

Now, the next three terms are

5^{th}

6^{th}

and

7^{th}

terms.

a_{5} = a + 4 d

= 3 + 4 \times \sqrt{2}

= 3 + 4 \sqrt{2}

a_{6} = a + 5 d

= 3 + 5 \times \sqrt{2}

= 3 + 5 \sqrt{2}

a_{7} = a + 6 d

= 3 + 6 \times \sqrt{2}

= 3 + 6 \sqrt{2}

Hence, the correct option is 2.

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