Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=40, r=\frac{1}{2} $$

Step-by-Step Solution

Answer

The first five terms of the sequence are 40, 20, 10, 5, and 2.5.

Step 1: Understand the Formula for Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by \(r\). The formula for the \(n^{th}\) term is \(a_{n} = a_{1} \times r^{(n-1)}\).

Step 2: Calculate the First Term

The first term of the sequence is already given as \(a_{1} = 40\).

Step 3: Calculate the Second Term

Use the formula for the \(n^{th}\) term: \(a_{2} = a_{1} \times r = 40 \times \frac{1}{2} = 20\).

Step 4: Calculate the Third Term

Continue using the formula: \(a_{3} = a_{2} \times r = 20 \times \frac{1}{2} = 10\).

Step 5: Calculate the Fourth Term

Find the next term: \(a_{4} = a_{3} \times r = 10 \times \frac{1}{2} = 5\).

Step 6: Calculate the Fifth Term

Finally, find the fifth term: \(a_{5} = a_{4} \times r = 5 \times \frac{1}{2} = 2.5\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the magic lies in the **common ratio**. It's what you multiply by every time you want the next number. If you know this value, you can almost "predict" every number in the sequence, like a well-told story. Think of it as the "rule" that keeps the sequence in check. Whether you're inflating a balloon or filling up a bucket, the common ratio keeps the rhythm steady.

• Given: The common ratio is often a fixed number.
• For our sequence: The common ratio (\(r\)) is \(\frac{1}{2}\).
• How it works: Multiply it by the last number you have!

In our example, the journey begins strong at 40, and each step forward scales everything by \(\frac{1}{2}\). This leads seamlessly into how the spiritual formula of the sequence operates.

The Role of the Sequence Formula

The sequence formula wraps up the entire charm of a geometric sequence in a neat package. Imagine having a secret recipe to find the exact term you want, whether it's the second, fifth, or hundredth.The formula is: \(a_{n} = a_{1} \times r^{(n-1)}\)

• \(n\): Your term number. Ask yourself, "Which one do I want?"
• \(a_{1}\): This is your starting number, the first term.
• \(r\): Our trusty common ratio.

Start with the first term, keep multiplying by the common ratio raised to the power of "one less than" the term number you want. It might sound technical, but once you see it in action, it’s as simple as pie! You saw it work perfectly for terms leading from \(a_1 = 40\) to \(a_5 = 2.5\).

Finding the First Term

The first term is like the VIP in our sequence story. It's the anchor that kicks off everything else. Without it, there's no starting point, no launch pad. In any geometric sequence, knowing the first term allows you to set the stage for every subsequent number.

• Stability: The sequence formula \(a_{n} = a_{1} \times r^{(n-1)}\) starts right here.
• For our exercise, we began with \(a_1 = 40\).
• Without this, all our calculations would fall into the world of guesswork!

Remember, the first term isn't just a number—it's the key that unlocks the entire geometric pattern! It logically places numbers in the sequence and interacts intricately with the common ratio to create the magic we observe.