Problem 63 - Algebra 2

Question

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=20, r=-0.5 $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_{n} = 20 \cdot (-0.5)^{n-1}\). The first three terms are \(20, -10, 5\).

Step 1: Determine the explicit formula

Substitute the given \(a_{1} = 20\) and \(r = -0.5\) into the explicit formula, resulting in \(a_{n} = 20 \cdot (-0.5)^{(n-1)}\).

Step 2: Generate the first three terms

Substitute \(n = 1, 2, 3\) into the found formula and calculate each term. For \(n = 1\), \(a_{1} = 20 \cdot (-0.5)^{(1-1)} = 20\). For \(n = 2\), \(a_{2} = 20 \cdot (-0.5)^{(2-1)} = -10\). For \(n = 3\), \(a_{3} = 20 \cdot (-0.5)^{(3-1)} = 5\). Thus, the first three terms are \(20, -10, 5\).

Key Concepts

Understanding the Explicit Formula

An explicit formula gives us a way to find any term directly in a geometric sequence without needing the previous term. It's a time saver because it allows us to jump directly to the term we want to calculate. In a geometric sequence, the explicit formula is expressed as:\[a_{n} = a_{1} imes r^{(n-1)}\]

• \(a_{n}\) is the nth term we're looking for.
• \(a_{1}\) is the first term of the sequence.
• \(r\) is the common ratio between terms.

In our exercise, knowing \(a_{1} = 20\) and \(r = -0.5\), we plug these values into the formula, resulting in \(a_{n} = 20 \times (-0.5)^{(n-1)}\). This formula helps us find any term with ease.

Decoding the Common Ratio

The common ratio is a key part of a geometric sequence, showing how one term relates to the next. It's a constant multiplier that tells you how to move from one term to the next:\[r = \frac{a_{2}}{a_{1}}\]Using the values given in the problem, \(r = -0.5\). This means each term is multiplied by \(-0.5\) to get the next one.

• If \(r\) is positive, the terms stay in the same sign.
• If \(r\) is negative, the terms alternate in sign.
• The magnitude of \(r\) tells you if the terms grow or shrink.

In our sequence, because \(r = -0.5\), the terms will alternate their signs and gradually decrease in size as we progress.

Finding Sequence Terms

Sequence terms are specific numbers that appear at different positions (n) in a sequence. By using the explicit formula, we can easily find any sequence terms:

• Term 1: Set \(n = 1\), \(a_{1} = 20\)
• Term 2: Set \(n = 2\), \(a_{2} = 20 \times (-0.5)^{1} = -10\)
• Term 3: Set \(n = 3\), \(a_{3} = 20 \times (-0.5)^{2} = 5\)

These calculations confirm the sequence starts with 20, then becomes -10, and then 5. By continuing this method, any subsequent terms in the sequence can be calculated easily.