Finding Terms In Geometric Progressions A Step By Step Guide

This article provides a comprehensive guide on how to find specific terms in geometric progressions. We will explore the fundamental concepts of geometric progressions and then delve into the step-by-step solutions for finding the 4th, 7th, and 9th terms ($T_4$, $T_7$, and $T_9$) for a variety of geometric sequences. Understanding geometric progressions is crucial in various fields, including mathematics, physics, finance, and computer science. A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Identifying the common ratio is the key to finding any term in the sequence.

The general form of a geometric progression is:

$a, ar, ar^2, ar^3, ar^4, ...$

where:

• $a$ is the first term,
• $r$ is the common ratio.

The nth term ($T_n$) of a geometric progression can be found using the formula:

$T_n = ar^{n-1}$

In this article, we will apply this formula to find the specified terms for different geometric progressions. To effectively apply the formula, it’s crucial to first identify the first term ($a$) and the common ratio ($r$) for each sequence. The common ratio can be found by dividing any term by its preceding term. Once we have these values, we can substitute them into the formula to calculate the desired terms. Now, let’s dive into the examples and solve them step-by-step to solidify your understanding of finding terms in geometric progressions.

(a) $3, 6, 12, 24, ...$

In this geometric progression, we first identify the first term and the common ratio. The first term ($a$) is clearly 3. To find the common ratio ($r$), we can divide any term by its preceding term. For example:

$r = \frac{6}{3} = 2$

$r = \frac{12}{6} = 2$

$r = \frac{24}{12} = 2$

Thus, the common ratio ($r$) is 2. Now we can use the formula $T_n = ar^{n-1}$ to find $T_4$, $T_7$, and $T_9$.

Finding $T_4$

To find the 4th term ($T_4$), we substitute $n = 4$, $a = 3$, and $r = 2$ into the formula:

$T_4 = 3 imes 2^{4-1} = 3 imes 2^3 = 3 imes 8 = 24$

So, the 4th term ($T_4$) is 24. This aligns with the given sequence, confirming our calculations.

Finding $T_7$

To find the 7th term ($T_7$), we substitute $n = 7$, $a = 3$, and $r = 2$ into the formula:

$T_7 = 3 imes 2^{7-1} = 3 imes 2^6 = 3 imes 64 = 192$

Thus, the 7th term ($T_7$) is 192.

Finding $T_9$

To find the 9th term ($T_9$), we substitute $n = 9$, $a = 3$, and $r = 2$ into the formula:

$T_9 = 3 imes 2^{9-1} = 3 imes 2^8 = 3 imes 256 = 768$

Therefore, the 9th term ($T_9$) is 768. By systematically applying the formula and identifying the first term and common ratio, we can accurately determine any term in the geometric progression.

(b) $5, 2\frac{1}{2}, 1\frac{1}{4}, \frac{5}{8}, ...$

For this geometric progression, we need to find the first term and the common ratio. The first term ($a$) is 5. The common ratio ($r$) can be found by dividing any term by its preceding term. First, let's convert the mixed numbers to improper fractions:

$2\frac{1}{2} = \frac{5}{2}$

$1\frac{1}{4} = \frac{5}{4}$

Now, we can find the common ratio:

$r = \frac{\frac{5}{2}}{5} = \frac{5}{2} imes \frac{1}{5} = \frac{1}{2}$

$r = \frac{\frac{5}{4}}{\frac{5}{2}} = \frac{5}{4} imes \frac{2}{5} = \frac{1}{2}$

Thus, the common ratio ($r$) is $\frac{1}{2}$. We can now use the formula $T_n = ar^{n-1}$ to find $T_4$, $T_7$, and $T_9$.

Finding $T_4$

To find the 4th term ($T_4$), we substitute $n = 4$, $a = 5$, and $r = \frac{1}{2}$ into the formula:

$T_4 = 5 imes (\frac{1}{2})^{4-1} = 5 imes (\frac{1}{2})^3 = 5 imes \frac{1}{8} = \frac{5}{8}$

So, the 4th term ($T_4$) is $\frac{5}{8}$, which matches the given sequence.

Finding $T_7$

To find the 7th term ($T_7$), we substitute $n = 7$, $a = 5$, and $r = \frac{1}{2}$ into the formula:

$T_7 = 5 imes (\frac{1}{2})^{7-1} = 5 imes (\frac{1}{2})^6 = 5 imes \frac{1}{64} = \frac{5}{64}$

Thus, the 7th term ($T_7$) is $\frac{5}{64}$.

Finding $T_9$

To find the 9th term ($T_9$), we substitute $n = 9$, $a = 5$, and $r = \frac{1}{2}$ into the formula:

$T_9 = 5 imes (\frac{1}{2})^{9-1} = 5 imes (\frac{1}{2})^8 = 5 imes \frac{1}{256} = \frac{5}{256}$

Therefore, the 9th term ($T_9$) is $\frac{5}{256}$. This example illustrates how to work with fractional common ratios in geometric progressions.

(c) $5, -10, 20, -40, 80$

In this geometric progression, we again identify the first term and the common ratio. The first term ($a$) is 5. The common ratio ($r$) can be found by dividing any term by its preceding term:

$r = \frac{-10}{5} = -2$

$r = \frac{20}{-10} = -2$

$r = \frac{-40}{20} = -2$

Thus, the common ratio ($r$) is -2. Now we can use the formula $T_n = ar^{n-1}$ to find $T_4$, $T_7$, and $T_9$.

Finding $T_4$

However, notice that the sequence is finite and has only 5 terms. $T_4$ is already given as -40. We can verify this using the formula:

$T_4 = 5 imes (-2)^{4-1} = 5 imes (-2)^3 = 5 imes -8 = -40$

So, the 4th term ($T_4$) is -40.

Finding $T_7$

To find the 7th term ($T_7$), we substitute $n = 7$, $a = 5$, and $r = -2$ into the formula:

$T_7 = 5 imes (-2)^{7-1} = 5 imes (-2)^6 = 5 imes 64 = 320$

Thus, the 7th term ($T_7$) is 320.

Finding $T_9$

To find the 9th term ($T_9$), we substitute $n = 9$, $a = 5$, and $r = -2$ into the formula:

$T_9 = 5 imes (-2)^{9-1} = 5 imes (-2)^8 = 5 imes 256 = 1280$

Therefore, the 9th term ($T_9$) is 1280. This example demonstrates how to handle negative common ratios, which result in alternating signs in the sequence.

(d) $1, -1\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, ...$

In this geometric progression, we identify the first term and the common ratio. The first term ($a$) is 1. Let's convert the mixed number to an improper fraction:

$-1\frac{1}{2} = -\frac{3}{2}$

Now, we can find the common ratio ($r$) by dividing any term by its preceding term:

$r = \frac{-\frac{3}{2}}{1} = -\frac{3}{2}$

$r = \frac{\frac{9}{4}}{-\frac{3}{2}} = \frac{9}{4} imes -\frac{2}{3} = -\frac{3}{2}$

$r = \frac{-\frac{27}{8}}{\frac{9}{4}} = -\frac{27}{8} imes \frac{4}{9} = -\frac{3}{2}$

Thus, the common ratio ($r$) is $-\frac{3}{2}$. Now we can use the formula $T_n = ar^{n-1}$ to find $T_4$, $T_7$, and $T_9$.

Finding $T_4$

$T_4$ is already given as $-\frac{27}{8}$. Let's verify it:

$T_4 = 1 imes (-\frac{3}{2})^{4-1} = (-\frac{3}{2})^3 = -\frac{27}{8}$

So, the 4th term ($T_4$) is indeed $-\frac{27}{8}$.

Finding $T_7$

To find the 7th term ($T_7$), we substitute $n = 7$, $a = 1$, and $r = -\frac{3}{2}$ into the formula:

$T_7 = 1 imes (-\frac{3}{2})^{7-1} = (-\frac{3}{2})^6 = \frac{729}{64}$

Thus, the 7th term ($T_7$) is $\frac{729}{64}$.

Finding $T_9$

To find the 9th term ($T_9$), we substitute $n = 9$, $a = 1$, and $r = -\frac{3}{2}$ into the formula:

$T_9 = 1 imes (-\frac{3}{2})^{9-1} = (-\frac{3}{2})^8 = \frac{6561}{256}$

Therefore, the 9th term ($T_9$) is $\frac{6561}{256}$. This final example reinforces the process of finding terms in geometric progressions, even with fractional and negative common ratios. In conclusion, to find specific terms in a geometric progression, first identify the first term ($a$) and the common ratio ($r$). Then, use the formula $T_n = ar^{n-1}$, substituting the values for $a$, $r$, and $n$ to find the desired term. This method applies regardless of whether the common ratio is an integer, a fraction, or a negative number.

In this comprehensive guide, we have meticulously demonstrated how to find specific terms in geometric progressions. By understanding the fundamental principles and applying the formula $T_n = ar^{n-1}$, one can easily calculate any term in a geometric sequence. Each example provided highlights different scenarios, such as integer common ratios, fractional common ratios, and negative common ratios, ensuring a thorough understanding of the concept. Whether you're a student learning about geometric progressions for the first time or someone looking to refresh your knowledge, this guide offers a clear and concise explanation of the process. The ability to work with geometric progressions is not only essential in mathematics but also has practical applications in various real-world scenarios. From calculating compound interest in finance to modeling population growth in biology, geometric sequences provide a powerful tool for understanding patterns and making predictions. Mastering the techniques outlined in this article will undoubtedly strengthen your mathematical skills and enhance your problem-solving abilities in various fields.