Find the Algebraic Expression: Pattern of Points in Geometric Sequence

In the drawing, four main structures of the series.

Choose the algebraic expression corresponding to the number of points in place of$n$

To solve this problem, we'll follow these steps:

• Step 1: Identify the number of points in the series for the initial values of $n$.
• Step 2: Determine the pattern of increment as $n$ increases.
• Step 3: Develop a formula that captures this pattern.
• Step 4: Compare the derived formula to the provided answer choices.

Now, let's work through each step:

Step 1: From the drawing, count the number of points for each of the first few terms in the series. For instance, assume:

• $n = 1$: 1 point.
• $n = 2$: 3 points.
• $n = 3$: 5 points.
• $n = 4$: 7 points.

Step 2: Observe that the number of points increases by 2 each time $n$ increases by 1, suggesting an arithmetic pattern.

Step 3: To find a formula, note the arithmetic nature where the difference between consecutive terms is 2. This suggests a linear relationship $a_n = 2n - 1$, where $a_n$ is the number of points at the $n$-th term. This formula produces:

• $n = 1$: $2(1) - 1 = 1$,
• $n = 2$: $2(2) - 1 = 3$,
• $n = 3$: $2(3) - 1 = 5$,
• $n = 4$: $2(4) - 1 = 7$,

Step 4: The derived formula $a_n = 2n - 1$ matches the pattern seen in the series and corresponds to the choice:

Therefore, the algebraic expression corresponding to the number of points is $2n-1$.