Problem 58 - Algebra 2

Question

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=1, r=2 $$

Step-by-Step Solution

Answer

The explicit formula for the geometric sequence is \(a_{n} = 2^{(n-1)}\). The first three terms of the sequence are 1, 2, 4.

Step 1: Insert the Given Values into the Formula

The given values are \(a_{1} = 1\) and \(r = 2\). The explicit formula for the sequence becomes \(a_{n} = 1 * 2^{(n-1)}\) which simplifies to \(a_{n} = 2^{(n-1)}\).

Step 2: Generate the First Three Terms

To find the first three terms of the sequence, plug in \(n = 1, 2, 3\) into the formula.\nFor \(n = 1\), we have \(a_{1}=2^{(1-1)}= 2^{0}=1\).For \(n = 2\), we have \(a_{2}=2^{(2-1)}=2^{1}=2\).For \(n = 3\), we have \(a_{3}=2^{(3-1)}=2^{2}=4\).So, the first three terms are \(1\), \(2\), and \(4\).

Key Concepts

Explicit Formula

In the world of geometric sequences, the explicit formula is the key tool to unlock any term you want to find. It provides a direct way of calculating each term based on its position, known as "n". Instead of calculating each term step-by-step, you can jump to any term directly.Let's break down the explicit formula: - It generally follows the form \[ a_n = a_1 \cdot r^{(n-1)} \] where - \(a_n\) is the term you're looking for, - \(a_1\) is the first term of the sequence, and - \(r\) is the common ratio.For our exercise, the explicit formula is slightly simplified because our first term \(a_1\) is 1. Therefore, our formula becomes \[ a_n = 1 \cdot 2^{(n-1)} = 2^{(n-1)} \]. Here, you're multiplying 1 with the common ratio raised to the power of the term's position minus one. Hence, this formula is straightforward to work with because every term is a power of 2.

First Three Terms

To understand the progression of the sequence, it's important to find the first few terms. This gives us a tangible sense of how the sequence grows. In our sequence:- The explicit formula is \[ a_n = 2^{(n-1)} \] To generate the first three terms, we plug \(n = 1, 2,\) and \(3\) into the formula.- For \(n = 1\): \[ a_1 = 2^{(1-1)} = 2^0 = 1 \] The sequence starts with 1. - For \(n = 2\): \[ a_2 = 2^{(2-1)} = 2^1 = 2 \] The next term doubles, so we have 2.- For \(n = 3\): \[ a_3 = 2^{(3-1)} = 2^2 = 4 \] Another doubling brings us to 4.These initial terms, 1, 2, and 4, showcase how quickly the sequence increases, following the pattern of powers of 2.

Common Ratio

The term "common ratio" is central to understanding geometric sequences. It defines how each term in the sequence relates to the previous one, setting the pattern of multiplication by the same number.- In our example, the common ratio, \(r\), is 2.- This means that each term is twice as large as the preceding term.In general, if \(r\) is greater than 1, the sequence will grow rapidly, displaying exponential growth. For example:- Start with a term: \(1\),- Multiply by the common ratio \(r = 2\);- You continue multiplying each subsequent term by 2.This repeated multiplication by the common ratio leads to the creation of a consistent and predictable pattern, making it easy to determine future terms and even unseen ones. By understanding this fundamental property, geometric sequences become easier to approach and solve.