How many integers are in the following list: 1, 2, 4, 8, ... , 2048?

Answer 1

The sequence 1, 2, 4, 8, ..., 2048 is a geometric sequence where a₁ = 1 and r= 2:

a_n = a₁·r^n-1

a_n = 2^n-1

We want a_n = 2048:

2048 = 2^n-1

2048 = 2ⁿ·2^-1 = 2ⁿ/2 [Exponent Property: 2^a+b = 2^a2^b]

4096 = 2ⁿ

If you know logarithms, you can use log base 2 and your calculator to find the answer. if you don't know logs yet, just try different values of n until you get the right one.

log₂(4096) = n

12 = n

2048 is the 12th term in the sequence, so the sequence has 12 terms in it.