Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Find the \(n\)th term, the fifth term, and the eighth term of the geometric sequence. $$2,2^{x+1}, 2^{2 x+1}, 2^{3 x+1}, \ldots$$

Step-by-Step Solution

Answer

\( a_n = 2^{nx-x+1} \); 5th term: \( 2^{4x+1} \); 8th term: \( 2^{7x+1} \).

Step 1: Identify the first term and common ratio

The first term of the sequence is given as the first number, which is 2. To find the common ratio, divide the second term by the first term. In this case, the second term is \(2^{x+1}\) and the first term is \(2\). The common ratio \( r \) is thus \( r = \frac{2^{x+1}}{2} = 2^x \).

Step 2: General formula for the nth term of a geometric sequence

The general formula for the \(n\)th term of a geometric sequence is given by \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio. For this sequence, \( a_1 = 2 \) and \( r = 2^x \). Thus, the \(n\)th term is \( a_n = 2 \cdot (2^x)^{n-1} = 2^{1 + (n-1)x} = 2^{nx - x + 1} \).

Step 3: Calculate the 5th term

Use the general formula \( a_n = 2^{nx - x + 1} \) to find the 5th term of the sequence by substituting \( n = 5 \). This gives us \( a_5 = 2^{5x - x + 1} = 2^{4x + 1} \).

Step 4: Calculate the 8th term

Similarly, substitute \( n = 8 \) into the general formula to find the 8th term. This yields \( a_8 = 2^{8x - x + 1} = 2^{7x + 1} \).

Step 5: Summarize the terms

The general \(n\)th term of the sequence is \( a_n = 2^{nx - x + 1} \). The 5th term is \( 2^{4x + 1} \), and the 8th term is \( 2^{7x + 1} \).

Key Concepts

nth term formula

In a geometric sequence, each term is found by multiplying the previous term by a constant value known as the common ratio. To find any term in the sequence, you need a formula that combines both the initial term and the common ratio. This is the nth term formula, defined as follows: \[ a_n = a_1 \cdot r^{n-1} \] where:

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

In our example, the first term \( a_1 \) is 2, and the common ratio \( r \) is \( 2^x \). So, the nth term formula becomes:\[ a_n = 2 \cdot (2^x)^{n-1} \]By simplifying, this leads to \[ a_n = 2^{nx - x + 1} \].You can use this formula to calculate any term of the sequence by simply plugging in the value of \( n \).

common ratio

The common ratio is a fundamental part of understanding geometric sequences. It is the factor by which we multiply to move from one term to the next. To determine the common ratio, divide any term in the sequence by the previous term.In our specific sequence starting with 2 and continuing with terms like \( 2^{x+1} \), the common ratio \( r \) can be calculated as:\[ r = \frac{2^{x+1}}{2} = 2^x \]This ratio \( 2^x \) is constant throughout the sequence, ensuring that each term is obtained by multiplying the previous term by \( 2^x \). Understanding this helps in generating the sequence and directly impacts the nth term formula.

exponent rules

Exponent rules are essential when working with geometric sequences that involve expressions with powers. In our sequence, each subsequent term involves powers of 2, often requiring simplification.Some important exponent rules include:

• \( a^m \cdot a^n = a^{m+n} \) – When you multiply powers with the same base, add the exponents.
• \( \frac{a^m}{a^n} = a^{m-n} \) – When you divide powers with the same base, subtract the exponents.
• \( (a^m)^n = a^{mn} \) – When raising a power to another power, multiply the exponents.

In our problem, the formula \( a_n = 2^{nx - x + 1} \) results from applying these rules, especially the power of a power rule, which simplifies \( (2^x)^{n-1} \) to \( 2^{(n-1)x} \).

sequence terms

Understanding sequence terms involves both identifying the specific terms and calculating them using the nth term formula. Let's take two specific terms as examples: the fifth and the eighth.Using the derived formula \( a_n = 2^{nx - x + 1} \), plug in:

• For the fifth term: \( n = 5 \). Substitute to get \( a_5 = 2^{4x + 1} \).
• For the eighth term: \( n = 8 \). Substitute to get \( a_8 = 2^{7x + 1} \).

These calculations show how the nth term formula helps find any term in a geometric sequence. It is essential to understand both the concept of the sequence and the practical use of the formula to calculate specific terms.