Geometric sequence

Question

How many geometric sequences of two or more positive integers begin with 1 and end with 1024?

Answer 1

Note that 1024 = 2¹⁰

The geometric sequences are like 1, ..., 2¹⁰

Common ratio r must be 2ⁿ where n is a factor of 10 so it will eventuall multiply up to 2¹⁰.

Possible geo sequences are:

1. 1, 2¹⁰ (r = 2¹⁰)
2. 1, 2⁵, 2¹⁰ (r = 2⁵)
3. 1, 2², 2⁴, 2⁶, 2⁸, 2¹⁰ (r= 2²)
4. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 (r =2)

Ans: Four geometric sequences satisfy the constraints.