Finding The First Term In Geometric Sequence A Step-by-Step Guide

Dive into the world of geometric sequences, where we unravel the mystery of finding the first term. Geometric sequences are a fascinating area of mathematics, characterized by a constant ratio between successive terms. In this article, we will explore a specific problem: determining the first term of a geometric sequence given certain information. We'll dissect the formula that governs these sequences and apply it to the data presented in a table. By the end of this journey, you'll not only know how to solve this particular problem but also gain a deeper understanding of geometric sequences and their properties.

Understanding geometric sequences is crucial not only for mathematical problem-solving but also for real-world applications. These sequences appear in various fields, from finance (compound interest) to physics (radioactive decay) and even computer science (algorithm analysis). The ability to identify and analyze geometric sequences can provide valuable insights and predictive power in diverse scenarios. In this context, the formula ${ a_n = a_1 imes r^{n-1} }$ acts as a cornerstone, allowing us to connect the terms of the sequence to their positions and the underlying ratio. Let’s start by understanding the fundamentals of geometric sequences.

In essence, a geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant. This constant is known as the common ratio, often denoted as ${ r }$. The first term, represented as ${ a_1 }$, sets the foundation for the entire sequence. From there, each subsequent term is generated by repeatedly multiplying by ${ r }$. For example, if the first term is 2 and the common ratio is 3, the sequence would be 2, 6, 18, 54, and so on. The general formula, ${ a_n = a_1 imes r^{n-1} }$, encapsulates this pattern, allowing us to calculate any term ${ a_n }$ in the sequence given the first term ${ a_1 }$, the common ratio ${ r }$, and the term's position ${ n }$. This formula is not just a mathematical construct; it’s a powerful tool for understanding and predicting the behavior of geometric sequences.

Deconstructing the Formula: ${ a_n = a_1 imes r^{n-1} }$

The formula ${ a_n = a_1 imes r^{n-1} }$ is the key to unlocking the secrets of geometric sequences. Let's break down each component to fully grasp its significance.

• ${ a_n }$: This represents the ${ n }$-th term of the sequence. It's the value we want to find at a specific position in the sequence. For example, if we want to find the 5th term, ${ n }$ would be 5, and ${ a_n }$ would be ${ a_5 }$.
• ${ a_1 }$: This is the first term of the sequence. It's the starting point from which all other terms are generated. Finding ${ a_1 }$ is often the objective in many geometric sequence problems.
• ${ r }$: This is the common ratio, the constant value by which each term is multiplied to get the next term. The common ratio is the defining characteristic of a geometric sequence. If ${ r }$ is greater than 1, the sequence grows exponentially; if it's between 0 and 1, the sequence decays.
• ${ n }$: This is the position or the term number in the sequence. It indicates which term we're referring to (e.g., 1st term, 5th term, 10th term).

Understanding how these components interact is crucial. The formula essentially states that any term in the sequence is equal to the first term multiplied by the common ratio raised to the power of one less than the term's position. The ${ n-1 }$ exponent reflects the fact that the first term (${ a_1 }$) is already given, and we only need to multiply by the ratio ${ n-1 }$ times to reach the ${ n }$-th term. This formula serves as a bridge connecting the first term, the common ratio, and any other term in the sequence, making it a fundamental tool for analyzing and solving problems related to geometric sequences.

Applying the Formula to the Table

Now, let's put our knowledge into action. We are given a table that provides information about a geometric sequence:

n	5	7
aₙ	567	5103

This table tells us that the 5th term (${ a_5 }$) is 567 and the 7th term (${ a_7 }$) is 5103. Our goal is to find the first term (${ a_1 }$). To achieve this, we'll use the formula ${ a_n = a_1 imes r^{n-1} }$ and the information provided.

We have two data points, which means we can set up two equations using the formula:

1. For ${ n = 5 }$ and ${ a_5 = 567 }$: ${ 567 = a_1 imes r^{5-1} }$ or ${ 567 = a_1 imes r^4 }$
2. For ${ n = 7 }$ and ${ a_7 = 5103 }$: ${ 5103 = a_1 imes r^{7-1} }$ or ${ 5103 = a_1 imes r^6 }$

Now we have a system of two equations with two unknowns (${ a_1 }$ and ${ r }$). To solve for ${ a_1 }$, we first need to find the common ratio ${ r }$. A strategic approach is to divide the second equation by the first equation. This will eliminate ${ a_1 }$ and allow us to solve for ${ r }$. Let's perform this step and see how it simplifies our problem.

Solving for the Common Ratio (r)

To find the common ratio ${ r }$, we'll divide the second equation by the first equation. This method is effective because it cancels out the ${ a_1 }$ term, leaving us with an equation solely in terms of ${ r }$. Here are the equations we established earlier:

1. ${ 567 = a_1 imes r^4 }$
2. ${ 5103 = a_1 imes r^6 }$

Dividing equation (2) by equation (1), we get:

${ \frac{5103}{567} = \frac{a_1 imes r^6}{a_1 imes r^4} }$

Notice that ${ a_1 }$ appears in both the numerator and the denominator, so it cancels out:

${ \frac{5103}{567} = r^2 }$

Now, we simplify the fraction on the left side:

${ 9 = r^2 }$

To solve for ${ r }$, we take the square root of both sides:

${ r = \pm \sqrt{9} }$

${ r = \pm 3 }$

This gives us two possible values for the common ratio: ${ r = 3 }$ or ${ r = -3 }$. Geometric sequences can have both positive and negative common ratios. A positive ratio indicates that the terms alternate in sign. Now that we have the possible values for ${ r }$, we can substitute them back into one of the original equations to solve for ${ a_1 }$. Let's choose the simpler equation, ${ 567 = a_1 imes r^4 }$, and substitute both values of ${ r }$ to find the corresponding values of ${ a_1 }$.

Finding the First Term (${ a_1 }$)

Now that we have two possible values for the common ratio, ${ r = 3 }$ and ${ r = -3 }$, we can substitute each of these into one of our original equations to solve for the first term, ${ a_1 }$. Let's use the equation ${ 567 = a_1 imes r^4 }$. This equation is relatively straightforward and will allow us to calculate ${ a_1 }$ efficiently.

Case 1: When ${ r = 3 }$

Substitute ${ r = 3 }$ into the equation:

${ 567 = a_1 imes (3)^4 }$

Calculate ${ 3^4 }$:

${ 567 = a_1 imes 81 }$

Now, divide both sides by 81 to isolate ${ a_1 }$:

${ a_1 = \frac{567}{81} }$

${ a_1 = 7 }$

So, when ${ r = 3 }$, the first term ${ a_1 }$ is 7.

Case 2: When ${ r = -3 }$

Substitute ${ r = -3 }$ into the equation:

${ 567 = a_1 imes (-3)^4 }$

Calculate ${ (-3)^4 }$. Since the exponent is even, the result will be positive:

${ 567 = a_1 imes 81 }$

Again, divide both sides by 81 to isolate ${ a_1 }$:

${ a_1 = \frac{567}{81} }$

${ a_1 = 7 }$

Interestingly, in both cases, we find that ${ a_1 = 7 }$. This result might seem surprising, but it highlights an important characteristic of geometric sequences. Regardless of whether the common ratio is positive or negative, the first term remains the same in this particular scenario. This is because raising a number to an even power results in a positive value, effectively negating the sign difference in the common ratio when calculating the 4th power. Therefore, the first term of the geometric sequence is 7.

Conclusion: The First Term Revealed

In this exploration of geometric sequences, we embarked on a mission to find the first term of a sequence given two of its terms. By dissecting the formula ${ a_n = a_1 imes r^{n-1} }$, we understood the relationship between terms, the common ratio, and their positions in the sequence. We successfully set up a system of equations, solved for the common ratio ${ r }$, and then used that value to determine the first term ${ a_1 }$.

Our journey revealed that the first term of the geometric sequence presented in the table is 7. This result underscores the power of the geometric sequence formula and the importance of understanding its components. Whether the common ratio is 3 or -3, the first term remains consistent, showcasing a unique property of these sequences. The ability to manipulate and apply this formula is invaluable in various mathematical contexts and real-world applications.

Geometric sequences are more than just a mathematical concept; they are a tool for understanding patterns and making predictions. From financial growth to physical decay, these sequences appear in diverse fields, making their study both practical and fascinating. By mastering the techniques presented in this article, you've equipped yourself with a valuable skill for tackling geometric sequence problems and appreciating the elegance of mathematical patterns.