Problem 25 - Precalculus Mathematics for Calculus

Question

Find the nth term of a sequence whose first several terms are given. $$2,4,8,16, \dots$$

Step-by-Step Solution

Answer

The nth term is \(a_n = 2^n\).

Step 1: Identify the Sequence Pattern

First, observe the given terms: 2, 4, 8, and 16. Notice that each term appears to be a pattern of multiplication. Specifically, each term is double the previous term.

Step 2: Formulate the Sequence Rule

With the identified pattern, assume the first term (\(a_1\)) is 2. Then, each subsequent term is multiplied by 2, meaning the sequence is geometric with a common ratio (\(r\)) of 2. Therefore, the nth term of the sequence can be expressed as: \(a_n = a_1 imes r^{(n-1)}\).

Step 3: Substitute Known Values

Substitute the known values into our formula where \(a_1 = 2\) and \(r = 2\). The formula simplifies to \(a_n = 2 imes 2^{(n-1)}\).

Step 4: Simplify the Expression

Further simplify the expression: since \(2 imes 2^{(n-1)} = 2^1 imes 2^{(n-1)}\), applying the property of exponents \(2^{1+(n-1)} = 2^n\). Thus, the nth term is \(a_n = 2^n\).

Key Concepts

nth term formula

The "nth term formula" is a powerful tool for finding any term in a sequence without having to list all preceding terms. In geometric sequences, this formula helps us directly calculate the value of the term at any position \( n \).
Identifying the rule or pattern that defines the sequence allows us to apply this formula. For a geometric sequence, the nth term formula is \( a_n = a_1 \times r^{(n-1)} \), where \( a_n \) is the nth term we want to find, \( a_1 \) is the first term of the sequence, and \( r \) is the common ratio.
Using this formula eliminates the need to calculate each previous term one by one, saving time and effort. It's essential to note both the first term and common ratio accurately, as these values are crucial parameters in our formula.

common ratio

In geometric sequences, the "common ratio" is a key factor that dictates the sequence's progression. It is the factor by which we multiply each term to get the subsequent term.
Identifying this ratio is fundamental. By examining the sequence \( 2, 4, 8, 16, \ldots \), we can see that each term is obtained by multiplying the previous term by 2. Therefore, our common ratio \( r \) is 2.

• The common ratio is a constant value;
• It can be positive or negative, though here it is positive.

Understanding which operations transform one term into the next can help determine the sequence's type, geometric in this case, and define that crucial constant for further calculations.

exponents

Exponents are vital in expressing repeated multiplication, especially apparent in geometric sequences like the one in this exercise. The nth term formula for geometric sequences uses exponents to denote successive multiplications of the common ratio.
When we write the nth term formula, \( a_n = a_1 \times r^{(n-1)} \), the exponent \((n-1)\) shows how many times \( r \) is multiplied by itself. For our sequence, each term is \( 2 \) raised to the power of \( n \), simplified as \( a_n = 2^n \).

• Exponents are a compact way to write the multiplication of the same number by itself;
• The properties of exponents allow simplifications like \( 2 \times 2^{n-1} = 2^n \).

Mastering how exponents work can greatly enhance your ability to tackle geometric sequences, simplifying complex multiplications into a clean exponential form.