Problem 21 - Algebra and Trigonometry

Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=5 ; \quad r=-1 $$

Step-by-Step Solution

Answer

The fifth term is 5, and the nth term is \( a_n = 5 \times (-1)^{(n-1)} \).

Step 1: Identify the formula for the nth term of a geometric sequence

The nth term of a geometric sequence can be found using the formula: \[ a_n = a_1 \times r^{(n-1)} \] where \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Substitute values to find the fifth term

Given \( a_1 = 5 \) and \( r = -1 \), substitute these values into the formula to find the fifth term (\( n = 5 \)): \[ a_5 = 5 \times (-1)^{(5-1)} \] Simplify the exponent: \[ a_5 = 5 \times (-1)^4 \] Since \((-1)^4 = 1\), we get: \[ a_5 = 5 \times 1 = 5 \]

Step 3: Find the nth term

Using the formula \( a_n = a_1 \times r^{(n-1)} \) and substituting \( a_1 = 5 \) and \( r = -1 \): \[ a_n = 5 \times (-1)^{(n-1)} \]

Key Concepts

nth term

To understand a geometric sequence, it's important to grasp the concept of the nth term. The nth term is simply the term at the position 'n' in the sequence. For instance, in our exercise, we were asked to find the fifth term, which is the position where n is 5. The nth term of any geometric sequence follows a specific formula:

\[ a_n = a_1 \times r^{(n-1)} \]

Here, \( a_n \) denotes the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. This formula is crucial because it helps us generate any term in the sequence without listing all the preceding terms. This efficiency is a big advantage in mathematics, especially for sequences with a large number of terms.

common ratio

The common ratio is what makes a sequence geometric. It is the constant factor between consecutive terms in the sequence. For example, in the sequence 5, -5, 5, -5,..., the common ratio (\( r \)) is -1 because each term is obtained by multiplying the previous term by -1.

Identifying the common ratio is simple:

• Take any term in the sequence (after the first one).
• Divide it by the term immediately before it.

For the sequence in our exercise: \( a_2 = -5 \ \frac{-5}{5} = -1 \)

Therefore, the common ratio is \( -1 \).Understanding this ratio is critical: it not only helps in generating further terms but also in recognizing the nature of the sequence (whether it will oscillate, grow, or decay).

exponentiation

Exponentiation plays a vital role in geometric sequences. In the formula for the nth term, \( r^{(n-1)} \), exponentiation indicates how many times to multiply the common ratio, \( r \), by itself.

Consider the fifth term from our example. We use the expression \((-1)^{(5-1)}\), which simplifies to \((-1)^4 \).

In this context:

• \( r \) is -1
• We raise it to the power of 4

Exponentiation rules help simplify this: \((-1)^4 = 1 \), because any negative number raised to an even power results in a positive number. This knowledge allows us to simplify calculations efficiently.

sequence formulas

Sequence formulas are pivotal in understanding and working with geometric sequences. The primary formula, as we used, is: \( a_n = a_1 \times r^{(n-1)} \). This formula facilitates finding the term at any position 'n' in the sequence.

Key components of this formula are:

• \( a_1 \): The first term
• \( r \): The common ratio
• \( n-1 \): The exponent that tells how many times to multiply the common ratio by itself.

For instance, the nth term formula helps find the fifth term in our exercise: \( a_5 = 5 \times (-1)^{(5-1)} \ a_5 = 5 \times 1 = 5 \). Sequence formulas allow us to understand the underlying patterns and efficiently calculate terms without manually writing out every single term. This is especially helpful for long sequences or when specific high-order terms are needed.