Problem 20 - 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. $$ a_{1}=-2 ; \quad r=4 $$

Step-by-Step Solution

Answer

The fifth term is \(-512\). The nth term is \(a_n = -2 \times 4^{n-1}\).

Step 1: Identify the Given Values

The first term of the geometric sequence is given as \(a_1 = -2\), and the common ratio is given as \(r = 4\).

Step 2: Formula for the nth Term of a Geometric Sequence

Recall the formula for the nth term of a geometric sequence: \[ a_n = a_1 \times r^{n-1} \]This formula will be used to find both the fifth term and the nth term in the sequence.

Step 3: Calculate the Fifth Term

To find the fifth term \(a_5\), plug in the values into the formula: \[ a_5 = (-2) \times 4^{5-1} \]Simplify the exponent: \[ a_5 = (-2) \times 4^4 \]Calculate \(4^4 = 256\): \[ a_5 = (-2) \times 256 = -512 \]So, the fifth term is \(-512\).

Step 4: Find the nth Term

Now, use the same formula for the nth term: \[ a_n = a_1 \times r^{n-1} \]Substitute \(a_1 = -2\) and \(r = 4\): \[ a_n = (-2) \times 4^{n-1} \]So, the nth term is \[ a_n = -2 \times 4^{n-1} \]

Key Concepts

nth term formula

In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio. This relationship can be expressed neatly in the formula for the nth term. The nth term formula helps you find any term in the sequence without listing all the terms. The formula is:
\[ a_n = a_1 \times r^{n-1} \]
Here, \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you're interested in. This formula is very powerful because it allows you to jump directly to any term in the sequence with minimal calculation.

common ratio

The common ratio (and ardefined as \ratio 4, 16, and 64 is 2.equation for any term.Hleys, equivalent for a detailed breakdown.\rhere are some key points:
The common ratio remains constant:

In the geometric sequence 2, 4, 8, 16, the common ratio is always 2 (each term is 2 times the previous one).

The common ratio can be positive or negative, and the terms will either keep increasing or oscillate.

To find the common ratio, divide any term by the previous term: \( r = \frac{a_{n}}{a_{n-1}} \)

Understanding and identifying the common ratio is crucial for using the nth term formula effectively.

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. This type of sequence grows (or reduces) exponentially. Here are some points to understand it better:

• In mathematical terms, if the first term is \( a_1 \) and the common ratio is \( r \), the sequence looks like: \( a_1, a_1r, a_1r^2, a_1r^3, ... \)
• Geometric sequences can model real-life phenomena, such as population growth, radioactive decay, and interest calculations.
• The general form of a geometric sequence is defined by the nth term formula: \( a_n = a_1 \times r^{n-1} \)
• For example, given \( a_1 = -2 \) and \( r = 4 \), the sequence is: \( -2, -8, -32, -128, ... \)

Understanding geometric progression helps in diverse fields such as finance, biology, and physics.