Question

The first figure of the Sierpinski triangle has one shaded triangle. The second figure of the Sierpinski triangle has three shaded triangles. The third figure of the Sierpinski triangle has nine shaded triangles. Which summation represents the total number of shaded triangles in the first 15 figures?

Sigma-Summation Underscript n = 1 Overscript 15 EndScripts 1 (3) Superscript n minus 1
Sigma-Summation Underscript n = 1 Overscript 15 EndScripts 3 (1) Superscript n minus 1
Sigma-Summation Underscript n = 1 Overscript 15 EndScripts 1 (one-third) Superscript n minus 1
Sigma-Summation Underscript n = 1 Overscript 15 EndScripts one-third (1) Superscript n minus 1

Answer 1

`\textsf{a)} \quad \displaystyle \sum^(15)_(n=1) 1(3)^(n-1)`

Step-by-step explanation:

The number of shaded triangles can be represented by a sequence:

• 1, 3, 9, ...

As each term in the sequence is 3 times the previous term, it is a geometric sequence with the common ratio of 3.

The general form of a geometric sequence is:

`a_n=ar^(n-1)`

where:

• 
  `a_n` is the nth term.
• 
  `a` is the first term.
• 
  `r` is the common ratio.
• 
  `n` is the position of the term.

Given the first term, a, is 1 and the common ratio, r, is 3, the equation to find the nth term of the sequence is:

`\boxed{a_n=1(3)^(n-1)}`

Sigma notation is a shorthand way of writing the sum of a sequence of terms. It is denoted by the Greek letter sigma (Σ) and is followed by the expression that describes the terms being summed.

Therefore, the summation that represents the total number of shaded triangles in the first 15 figures is:

`\boxed{\displaystyle \sum^(15)_(n=1) 1(3)^(n-1)}`