Which description best represents the graph of a geometric sequence $a{r}_n$ with $a\ne 0$ and $r>1$ when plotted as discrete points $(n,a_n)$?

• A. A set of points that lie on a straight line with constant slope (linear growth)
• B. A set of points that increase by a constant ratio, curving upward so the y-values grow faster as $n$ increases (exponential growth)
• C. A set of points forming a parabola symmetric about a vertical line (quadratic growth)
• D. A set of points that oscillate periodically above and below the x-axis (sinusoidal behavior)

Answer: (B)

Verified step by step guidance

1. Recall that a geometric sequence is defined by the formula \(a_n = a \cdot r^n\), where \(a\) is the initial term and \(r\) is the common ratio.
2. Given that \(a \neq 0\) and \(r > 1\), each term is obtained by multiplying the previous term by a number greater than 1, which means the terms grow larger as \(n\) increases.
3. When plotting the points \((n, a_n)\), since \(a_n\) grows by a constant ratio rather than a constant difference, the points will not lie on a straight line (which would indicate linear growth).
4. Instead, the points will form a curve that increases more and more steeply, showing exponential growth because the value of \(a_n\) increases faster as \(n\) gets larger.
5. Therefore, the graph is best described as a set of points that increase by a constant ratio, curving upward so the y-values grow faster as \(n\) increases.