Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=2^{n-1}$$

Step-by-Step Solution

Answer

1, 2, 4, 8, 16.

Step 1: Identify the Sequence Formula

The formula for the sequence provided is \( a_n = 2^{n-1} \). This means each term is obtained by raising 2 to the power of \( n-1 \).

Step 2: Calculate the First Term

To find the first term \( a_1 \), substitute \( n = 1 \) into the formula: \( a_1 = 2^{1-1} = 2^0 = 1 \).

Step 3: Calculate the Second Term

To find the second term \( a_2 \), substitute \( n = 2 \) into the formula: \( a_2 = 2^{2-1} = 2^1 = 2 \).

Step 4: Calculate the Third Term

To find the third term \( a_3 \), substitute \( n = 3 \) into the formula: \( a_3 = 2^{3-1} = 2^2 = 4 \).

Step 5: Calculate the Fourth Term

To find the fourth term \( a_4 \), substitute \( n = 4 \) into the formula: \( a_4 = 2^{4-1} = 2^3 = 8 \).

Step 6: Calculate the Fifth Term

To find the fifth term \( a_5 \), substitute \( n = 5 \) into the formula: \( a_5 = 2^{5-1} = 2^4 = 16 \).

Step 7: List the First Five Terms

The first five terms of the sequence are 1, 2, 4, 8, and 16.

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that represents elements of a sequence in a logical order. In the context of arithmetic sequences, the sequence formula determines how we calculate each term. For the exercise given, the sequence formula is \( a_n = 2^{n-1} \). This indicates the term \( a_n \) is obtained by raising 2 to the power of \( n-1 \). Understanding this conceptual framework makes it much easier to calculate each term without any tools like calculators.

• Each sequence term depends on its position \( n \).
• Substitute the value of \( n \) into the formula to find the corresponding term.
• This particular sequence grows exponentially, as each subsequent term is a power of 2.

With this information, we can confidently compute any number within the sequence by simply plugging the desired \( n \) value into the formula.

Exponents

Exponents are a crucial part of mathematics that simplify long multiplication processes. When we talk about exponents, we mean repeatedly multiplying a number by itself. In this exercise, the number 2 is being raised to the power of \( n-1 \). For example, raising 2 to the power of 3 is written as \( 2^3 \), which equals 8 because it is equivalent to multiplying 2 by itself three times (\( 2 \times 2 \times 2 = 8 \)).
Exponents have several important properties that make calculations simpler:

• **Power of Zero**: Any number raised to the power of zero is 1. For example, \( 2^0 = 1 \).
• **Power of One**: Any number raised to the power of one is the number itself. Hence, \( 2^1 = 2 \).
• **Higher Powers**: As the exponent increases, the result grows rapidly due to repeated multiplication.

Mastering exponents is key when working with sequence formulas like \( a_n = 2^{n-1} \), as they dictate the process of calculating each term's value.

Term Calculation

Calculating the terms in a sequence can initially seem daunting, but with a clear process, it becomes straightforward. Let's delve into how term calculation was performed in the provided exercise. To find a term's value, \( n \) is substituted into the sequence formula \( a_n = 2^{n-1} \). This simple substitution allows us to easily calculate any term:

• **First Term**: \( n = 1 \). Thus, \( a_1 = 2^{1-1} = 2^0 = 1 \).
• **Second Term**: \( n = 2 \). So, \( a_2 = 2^{2-1} = 2^1 = 2 \).
• **Third Term**: \( n = 3 \). Therefore, \( a_3 = 2^{3-1} = 2^2 = 4 \).
• **Fourth Term**: \( n = 4 \). Thus, \( a_4 = 2^{4-1} = 2^3 = 8 \).
• **Fifth Term**: \( n = 5 \). So, \( a_5 = 2^{5-1} = 2^4 = 16 \).

With practice, term calculation using sequence formulas becomes an easy task. Remember, it's all about substituting \( n \) into the formula and calculating the exponent to get the term's value.