Natasha has to determain the number of toothpicks in the 15th step, step one has four toothpicks step two has 12 toothpicks and step three has 24 toothpicks. She does not want to draw 15 pictures then count the toothpicks. Explain or show how she could do this.

tn = a1 + (n-1)d

t2 = 12 and t3 = 24 so d= 24-13=11

so we need to find a1

t2=a1 + (n-1)d
12 = a1 + (2-1)11
12 = a1 +11
1 = a1

then

t15 = 1 + (15-1)11
t15 = 1 + 154
t15 = 155

use the arithmetic sequence formula

a(n)=a(1)+(n-1)d
last=first+(number-1)difference
?=4+(15-1)20
?=284

im pretty sure lol

To determine the number of toothpicks in the 15th step without drawing 15 pictures, Natasha can use patterns to find a formula or equation that represents the relationship between the step number and the number of toothpicks.

By looking at the given information, we can observe a clear pattern:

- Step 1 has 4 toothpicks.
- Step 2 has three times the number of toothpicks in the previous step (3 * 4 = 12).
- Step 3 has twice the number of toothpicks in the previous step (2 * 12 = 24).

From these patterns, we can infer that:
- The number of toothpicks in each step is increasing by a factor of 3.
- Each step is adding an additional 8 toothpicks compared to the previous step.

Now, let's continue using these observations to derive a general formula for finding the number of toothpicks in any given step.

Step 1: 4 toothpicks
Step 2: 4 + 3 * 8 = 4 + 24 = 28 toothpicks
Step 3: 28 + 3 * 8 = 28 + 24 = 52 toothpicks

By following this pattern, we can establish the formula for finding the number of toothpicks in any given step:
Number of toothpicks = 4 + (step number - 1) * 8

Let's plug in the value of the 15th step using this formula:
Number of toothpicks = 4 + (15 - 1) * 8
Number of toothpicks = 4 + 14 * 8
Number of toothpicks = 4 + 112
Number of toothpicks = 116

Therefore, in the 15th step, there will be 116 toothpicks without the need to draw 15 pictures and count each one.