If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then ap, bq, cr are in

What is the sum of n terms of the series $$\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + ...$$?

What is the greatest value of the positive integer n satisfying the condition?

$$1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + .... + {1 \over {{2^{n - 1}}}} < 2 - {1 \over {1000}}$$

How many geometric progressions is/are possible containing 27, 8 and 12 as three of its/their terms?