A simple fractal tree Grows in stages. A each new stage, two new line segments branch out from each segment at the top of the tree. The first five stages are shown. How many line segments need to be drawn to create stage 20?

Stage 1- 1
Stage 2 - 3
Stage 3 - 7
Stage 4 -15
Stage 5 - 31 branches

(2^n) - 1

To determine the number of line segments needed to create stage 20 of the fractal tree, we can observe a pattern.

At each stage, the number of line segments doubles and then adds one.

Starting from stage 1, the number of line segments is:

Stage 1 - 1
Stage 2 - 2 * 1 + 1 = 3
Stage 3 - 2 * 3 + 1 = 7
Stage 4 - 2 * 7 + 1 = 15
Stage 5 - 2 * 15 + 1 = 31

We can see that at each stage, the number of line segments is equal to twice the number of line segments in the previous stage plus one.

Using this pattern, we can find the number of line segments for stage 20.

Stage 20 = 2 * (Stage 19) + 1
= 2 * (2 * (Stage 18) + 1) + 1
= 2 * (2 * (2 * (Stage 17) + 1) + 1) + 1
= ...
= 2^19 + 2^18 + 2^17 + ... + 2 + 1

We can use the formula for the sum of a geometric series to find the sum of the powers of 2:

Sum = (first term * (1 - common ratio ^ number of terms)) / (1 - common ratio)

In this case, the first term is 1, the common ratio is 2, and the number of terms is 20.

Sum = (1 * (1 - 2^20)) / (1 - 2)
= (1 * (1 - 2^20)) / (-1)
= (2^20 - 1) / (-1)
= 2^20 - 1

Therefore, the number of line segments needed to create stage 20 of the fractal tree is 2^20 - 1, which is 1,048,575.

To find out how many line segments need to be drawn to create stage 20 of the fractal tree, we can observe a pattern in the number of branches for each stage.

From stage 1 to stage 2, the number of branches doubles. (1 to 3)
From stage 2 to stage 3, the number of branches doubles. (3 to 7)
From stage 3 to stage 4, the number of branches doubles. (7 to 15)
From stage 4 to stage 5, the number of branches doubles. (15 to 31)

We can see that the pattern is that the number of branches doubles with each stage.

To calculate the number of branches at any given stage, we can use the formula:
Number of branches = 2^(stage - 1)

Using this formula, we can find the number of branches at stage 20:
Number of branches = 2^(20 - 1)
Number of branches = 2^19
Number of branches = 524,288

Therefore, to create stage 20 of the fractal tree, we would need to draw 524,288 line segments.

Does the sequence look anything like powers of 2?

2 4 8 16 32 ...