Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the \(n\)th Term of a Geometric Sequence Write the first five terms of the geometric sequence. Find the common ratio and write the \(n\)th term of the sequence as a function of \(n.\) $$a_{1}=30, a_{k+1}=-\frac{2}{3} a_{k}$$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are 30, -20, \frac{40}{3}, -\frac{80}{9}, and \frac{160}{27}. The common ratio is -\frac{2}{3}. The nth term of the sequence is given by the expression \(a_{n}=30\left(-\frac{2}{3}\right)^{n-1}\).

Step 1: Find the first five terms

Start with the provided first term, \(a_{1}=30\). Then use the given formula \(a_{k+1}=-\frac{2}{3} a_{k}\) to calculate the next four terms of the sequence. Using this recipe: \n\(a_{2}=-\frac{2}{3}a_{1}\), \n\(a_{3}=-\frac{2}{3}a_{2}\), \n\(a_{4}=-\frac{2}{3}a_{3}\), and \n\(a_{5}=-\frac{2}{3}a_{4}\).

Step 2: Identify the common ratio

The common ratio is found by dividing any term by the previous term in the sequence. You can pick any of the calculated terms and divide by its predecessor to find the ratio. For example, \(r=\frac{a_{2}}{a_{1}}= -\frac{2}{3}\).

Step 3: Write the nth term

Now that we have the first term and the common ratio, we can write an expression for the nth term of the sequence using the formula \(a_{n}=a_{1}r^{n-1}\). Substituting \(a_{1}=30\) and \(r=-\frac{2}{3}\) into this formula gives the expression for the nth term: \(a_{n}=30\left(-\frac{2}{3}\right)^{n-1}\).

Key Concepts

Common Ratio

Ever noticed how certain patterns shrink or grow steadily? A similar concept exists in mathematics, known as the common ratio in geometric sequences. Think of it like a consistent