Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence. $$a_{1}=4, a_{4}=\frac{1}{2}, 10 \text { th term }$$

Step-by-Step Solution

Answer

The 10th term of the geometric sequence is \(1/128\).

Step 1: Find the common ratio relationship

The same ratio relationship exists between the term in the fourth place and the first, and the term in the tenth place and the fourth. Therefore, the ratio \(r\) can be calculated as \(r = (a_4 / a_1)^(1/3)\). This is because in the geometric sequence, the 4th term would be \(a_1\) multiplied by \(r^3\). Hence, we isolate \(r\). Note that we calculate the cube root because three terms are moving from \(a_1\) to \(a_4\) in the sequence.

Step 2: Calculate the common ratio

Now, substitute the given values into the equation from Step 1 to find the value of \(r\). Therefore, \(r = (1/2 / 4)^(1/3) = 1/2\).

Step 3: Substitute into the formula for the Nth term of a geometric sequence

We know \(a_1\) (4) and \(r\) (1/2). Substitute these numbers into the formula for the nth term of a geometric sequence \(a_n = a_1 \cdot r^{(n-1)}\). In this case, we are looking for the 10th term of the sequence, so \(n = 10\). Substitute into the formula to get \(a_{10} = 4 \cdot (1/2)^{10-1}\).

Step 4: Calculate the 10th term

Finally, compute the 10th term as follows. \(a_{10} = 4 \cdot (1/2)^9 = 4 \cdot 1/512 = 1/128\).

Key Concepts

Common Ratio

The common ratio in a geometric progression is the factor by which each term is multiplied to obtain the subsequent term. Consider having a series of numbers where each number is a certain multiple of the one before it. That multiple is known as the common ratio, represented by the variable 'r'.

For example, if a sequence starts with 1 and each subsequent term is triple the previous one, the sequence would be 1, 3, 9, 27, and so on. Here, the common ratio 'r' is 3, because each term is three times greater than the previous term.In the given exercise, we need to find this multiplier that links every two consecutive terms in the sequence. The significance of identifying the common ratio is paramount, as it lays the foundation for predicting subsequent terms in the series and for a broader understanding of the sequence's behavior.

Geometric Progression

A geometric progression (also known as geometric sequence) is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The progression is described by its first term, designated as \(a_1\), and the common ratio, 'r'.

When visualizing geometric progression, think of it like a chain reaction, where each term generates the next one in a predictable manner, based on the common ratio. This quality of the geometric sequence makes it a powerful tool in various fields such as banking, physics, and computer science, wherever exponential growth or decay is involved.In our textbook problem, the goal is to discover a specific term in such a progression. Understanding the relationship between the terms and the common ratio is crucial to solving the problem, making the concept of geometric progression indispensable.

Nth Term Formula

The nth term formula is the key to unraveling the mystery of any term in a geometric series. It is mathematically represented as \(a_n = a_1 \times r^{(n-1)}\), where \(a_n\) is the term you are looking for, \(a_1\) is the first term of the sequence, 'r' is the common ratio, and 'n' stands for the term's position in the sequence.

This formula serves as a bridge, allowing you to leap directly to any term in the sequence without tediously multiplying the common ratio again and again. It's incredibly powerful for both short sequences and astronomically large ones where manual calculations would be impractical and time-consuming. For our exercise, we neatly plug in the values for \(a_1\), 'r', and 'n' to compute \(a_{10}\), the tenth term, exemplifying how the nth term formula turns a potentially complex process into a simple substitution and calculation.