Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=1, r=\frac{1}{2}$$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are: 1, \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\)

Step 1: Write down the first term

The first term \(a_{1}\) of the series is given as 1, so it is known already: \(a_{1}=1\)

Step 2: Calculate the second term

The second term can be found by multiplying the first term by the ratio \(r\). This gives: \(a_{2}=r \cdot a_{1} = \frac{1}{2} \cdot 1 = \frac{1}{2}\).

Step 3: Calculate the third term

Identically, the third term is found by multiplying the second term by the ratio \(r\). This gives: \(a_{3} = r \cdot a_{2} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\).

Step 4: Calculate the fourth term

Similarly, the fourth term is found by multiplying the third term by the ratio \(r\). This gives: \(a_{4} = r \cdot a_{3} = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}\).

Step 5: Calculate the fifth term

Lastly, the fifth term is found by multiplying the fourth term by the ratio \(r\). This gives: \(a_{5} = r \cdot a_{4} = \frac{1}{2} \cdot \frac{1}{8} = \frac{1}{16}\).

Key Concepts

Geometric Progression

A geometric progression, or 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. It differs from an arithmetic sequence where terms increase by adding a constant. Instead, in a geometric sequence, you multiply by a constant. This repetitive multiplication creates a pattern seen in the sequence.

• Example: If the first term is 1 and the common ratio is \( \frac{1}{2} \), the sequence is 1, \( \frac{1}{2} \), \( \frac{1}{4} \), \( \frac{1}{8} \), \( \frac{1}{16} \).
• Common examples of geometric progressions in real life include interest rates, exponential growth, and radioactive decay.

Breaking down the sequence into these simple parts makes it easier to understand the concept as a whole. Remember that the first term is always given, and the nature of multiplication makes this sequence grow rapidly or decay depending on whether the common ratio is greater than or less than one.

Common Ratio

The common ratio in a geometric progression is the key factor that dictates the sequence's nature. It is denoted by \(r\) and can be any real number, except zero.

• If \( r = 1 \), the sequence becomes constant as each term equals the first.
• If \( r > 1 \), the sequence grows exponentially, increasing larger with each term.
• If \( 0 < r < 1 \), the terms of the sequence decrease, growing closer to zero.
• If \( r < 0 \), the sequence alternatively switches between positive and negative, adding complexity.

Understanding the common ratio's effect is crucial; it helps you predict and manipulate the progression of terms. In the original problem, with \( r = \frac{1}{2} \), each term halves the last, illustrating a shrinking sequence.

Term Calculation

Calculating terms in a geometric sequence involves a simple formula: For the \(n\)-th term \(a_n\), the formula is \(a_n = a_1 \cdot r^{n-1}\).

• Start with the first term, \(a_1\).
• Multiply by the common ratio \(r\) raised to the power of \(n-1\) for the desired term \(n\).

For example, the original sequence starts with \(a_1 = 1\) and \(r = \frac{1}{2}\):

• Second term: \(a_2 = 1 \cdot \left(\frac{1}{2}\right)^{2-1} = \frac{1}{2}\).
• Third term: \(a_3 = 1 \cdot \left(\frac{1}{2}\right)^{3-1} = \frac{1}{4}\).
• Continuing similarly finds all terms.

This method is straightforward once you grasp the concept of exponentiation and the regularity of multiplication in sequences. It allows prediction without sequentially following each prior term, especially helpful for large sequences.