Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$\frac{1}{3},-\frac{2}{9}, \frac{4}{27},-\frac{8}{81}, \dots$$

Step-by-Step Solution

Answer

The nth term of the sequence is \( \frac{(-2)^{n-1}}{3^n} \).

Step 1: Identify the first term and common ratio

The first term of the sequence is \( \frac{1}{3} \) and the common ratio is \( -\frac{2}{3} \) as each term is \( -\frac{2}{3} \) times the preceding term.

Step 2: Write the expression

The nth term of a geometric sequence can be found by multiplying the first term by the common ratio raised to the power of \( n - 1 \). The final expression becomes \( \frac{1}{3} * \left(-\frac{2}{3}\right)^{n-1} \).

Step 3: Simplify the expression

The expression can be simplified to \( \frac{(-2)^{n-1}}{3^n} \). This is the nth term of the sequence.

Key Concepts

nth Term Formula

Geometric sequences can seem daunting at first, but once you understand the nth term formula, it becomes much simpler. The nth term formula is used to find any term in a geometric sequence without having to list all prior terms. Think of it like a shortcut!In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. We use the nth term formula: \[ a_n = a_1 imes r^{n-1} \] Here's what it means:

• \( a_n \) is the term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number you're trying to find.

This formula is fantastic because it bypasses the need to calculate every prior term, saving you time — especially useful in large sequences.

Common Ratio

Understanding the common ratio is key when dealing with geometric sequences. But what is it exactly?The common ratio is the constant factor between consecutive terms of a geometric sequence. You find it by dividing any term by the one before it. For our specific sequence:First term: \( \frac{1}{3} \) Second term: \( -\frac{2}{9} \) To find the common ratio \( r \), divide the second term by the first term:\[ r = \frac{-\frac{2}{9}}{\frac{1}{3}} = -\frac{2}{3} \]This consistent multiplier makes geometric sequences both predictable and easy to work with once you know this magic number. Each term is simply the previous one multiplied by \( -\frac{2}{3} \), following the sequence's unique pattern.

Sequence Simplification

Simplifying expressions in geometric sequences can make them much easier to work with. Let's take the nth term expression from our example:\[ a_n = \frac{1}{3} \times \left(-\frac{2}{3}\right)^{n-1} \]Here's how we simplify it:

• Notice the pattern in the powers and denominators.
• Recognize that every term is a fraction where the numerator grows by a power of \(-2\).
• The denominator is the power of 3, increasing as n does.

This simplifies to:\[ a_n = \frac{(-2)^{n-1}}{3^n} \]Breaking it down, the numerator \((-2)^{n-1}\) changes signs due to the negative sign in the common ratio, making alternating terms positive and negative. Meanwhile, the \(3^n\) in the denominator shows how 3 multiplies with itself across terms. Thus, creating an efficient, clean expression that reveals the sequence's inherent structure!