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?

Read the following information carefully and answer the questions given below.

Let a, x, y, z, b be in AP, where x + y + z = 15. Let a, p, q, r, b be in HP, where p^$$-$$1 + q^$$-$$1 + r^$$-$$1 = 5/3.

What is the value of ab?

Read the following information carefully and answer the questions given below.

Let a, x, y, z, b be in AP, where x + y + z = 15. Let a, p, q, r, b be in HP, where p^$$-$$1 + q^$$-$$1 + r^$$-$$1 = 5/3.

What is the value of xyz?