Question

Match the graphs with the corresponding formulas

1.s_n = 4 + 1/n
2.s_n = 4 + (-1)^n/n
3.s_n = 4 - 1/n
4.s_n = 4/n

Answer 1

The corresponding formulas for the given graphs are:

1. \(s_n = 4 + \frac{1}{n}\)

2. \(s_n = 4 + \frac{(-1)^n}{n}\)

3. \(s_n = 4 - \frac{1}{n}\)

4. \(s_n = \frac{4}{n}\)

Step-by-step explanation:

Let's analyze each formula to match them with their respective graphs.

Formula 1 (\(s_n = 4 + \frac{1}{n}\)):

This formula indicates a sequence where \(s_n\) increases as \(n\) grows. As \(n\) increases, \(\frac{1}{n}\) tends towards zero, causing \(s_n\) to grow. This aligns with the graph of a curve ascending as \(n\) increases.

Formula 2 (\(s_n = 4 + \frac{(-1)^n}{n}\)):

Here, the sequence alternates between positive and negative values based on the parity of \(n\). When \(n\) is even, \(\frac{(-1)^n}{n}\) is positive, and when \(n\) is odd, it's negative. This behavior creates an oscillating graph.

Formula 3 (\(s_n = 4 - \frac{1}{n}\)):

This formula also shows an increasing sequence, but the rate of increase is slower than in Formula 1 due to the subtracted term \(\frac{1}{n}\). Consequently, the graph rises but at a gentler rate.

Formula 4 (\(s_n = \frac{4}{n}\)):

This sequence decreases as \(n\) grows because \(\frac{4}{n}\) tends towards zero as \(n\) increases. This decreasing behavior matches with a graph that descends as \(n\) grows.

In summary, by analyzing the behavior of each formula in relation to how they change with \(n\), we can match them accurately with their corresponding graphs based on their distinct characteristics."