jmc

Question

The sequence 1, 2, 4, 5, 10, 11, 22, 23, 46, 47,

\dots

is formed as follows:

∙

Start with the number 1.

∙

Add one to get 2.

∙

Double that to get 4.

∙

Add one to get 5.

∙

Double that to get 10, and so on.

We repeat the steps of "add one" and "double that", alternating between them.

The 100th term will be of the form

3 \cdot 2^{k} - 1.

Compute

k .

Accepted Answer

If we take every other term, starting with the second term 2, we get

2, 5, 11, 23, 47, \dots .

If we add one to each of these terms, we get

3, 6, 12, 24, 48, \dots .

Each term appear to be double the previous term.

To confirm this, let one term in the original sequence be

x - 1,

after we have added 1. Then the next term is

2 (x - 1) = 2 x - 2,

and the next term after that is

2 x - 2 + 1 = 2 x - 1.

This confirms that in the sequence 3, 6, 12, 24, 48,

\dots,

each term is double the previous term. Then the 50th term in this geometric sequence is

3 \cdot 2^{49},

so the 100th term in the original sequence is

3 \cdot 2^{49} - 1,

so

k = 49 .