Terms Of A Geometric Sequence Are Found By The Formula T N = A R N − 1 T_n = A R^{n-1} T N ​ = A R N − 1 . If A = 3 A = 3 A = 3 And R = 2 R = 2 R = 2 , Find The First Few Terms Of The Sequence.

Introduction

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula for the nth term of a geometric sequence is given by $T_n = a r^{n-1}$, where $a$ is the first term and $r$ is the common ratio. In this article, we will explore the formula for a geometric sequence and use it to find the first few terms of a sequence with $a = 3$ and $r = 2$.

Understanding the Formula

The formula for the nth term of a geometric sequence is $T_n = a r^{n-1}$. This formula tells us that to find the nth term of the sequence, we need to multiply the first term $a$ by the common ratio $r$ raised to the power of $n-1$. For example, if we want to find the 5th term of the sequence, we would use the formula $T_5 = a r^{5-1} = a r^4$.

Finding the First Few Terms

Now that we have a good understanding of the formula, let's use it to find the first few terms of the sequence with $a = 3$ and $r = 2$. We will start by finding the first few terms of the sequence and then use them to find the next few terms.

First Term

The first term of the sequence is given by $T_1 = a r^{1-1} = a r^0 = a$. Since $a = 3$, the first term of the sequence is $T_1 = 3$.

Second Term

The second term of the sequence is given by $T_2 = a r^{2-1} = a r^1 = a r$. Since $a = 3$ and $r = 2$, the second term of the sequence is $T_2 = 3 \cdot 2 = 6$.

Third Term

The third term of the sequence is given by $T_3 = a r^{3-1} = a r^2 = a r^2$. Since $a = 3$ and $r = 2$, the third term of the sequence is $T_3 = 3 \cdot 2^2 = 3 \cdot 4 = 12$.

Fourth Term

The fourth term of the sequence is given by $T_4 = a r^{4-1} = a r^3 = a r^3$. Since $a = 3$ and $r = 2$, the fourth term of the sequence is $T_4 = 3 \cdot 2^3 = 3 \cdot 8 = 24$.

Fifth Term

The fifth term of the sequence is given by $T_5 = a r^{5-1} = a r^4 = a r^4$. Since $a = 3$ and $r = 2$, the fifth term of the sequence is $T_5 = 3 \cdot 2^4 = 3 \cdot 16 = 48$.

Conclusion

Introduction

In our previous article, we explored the formula for a geometric sequence and used it to find the first few terms of a sequence with $a = 3$ and $r = 2$. In this article, we will answer some common questions about geometric sequences and provide additional information to help you better understand this topic.

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the formula for a geometric sequence?

A: The formula for the nth term of a geometric sequence is $T_n = a r^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

Q: How do I find the first few terms of a geometric sequence?

A: To find the first few terms of a geometric sequence, you can use the formula $T_n = a r^{n-1}$ and substitute the values of $a$ and $r$ into the formula. For example, if you want to find the first few terms of a sequence with $a = 3$ and $r = 2$, you would use the formula $T_n = 3 \cdot 2^{n-1}$.

Q: What is the common ratio?

A: The common ratio is a fixed, non-zero number that is used to find each term of a geometric sequence. It is denoted by the letter $r$ and is used in the formula $T_n = a r^{n-1}$.

Q: How do I find the common ratio?

A: To find the common ratio, you can use the formula $r = \frac{T_{n+1}}{T_n}$, where $T_{n+1}$ is the next term in the sequence and $T_n$ is the current term.

Q: What is the sum of a geometric sequence?

A: The sum of a geometric sequence is given by the formula $S_n = \frac{a(1-r^n)}{1-r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I find the sum of a geometric sequence?

A: To find the sum of a geometric sequence, you can use the formula $S_n = \frac{a(1-r^n)}{1-r}$ and substitute the values of $a$, $r$, and $n$ into the formula.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic sequence, on the other hand, is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

Conclusion

In this article, we have answered some common questions about geometric sequences and provided additional information to help you better understand this topic. We have seen that a geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, and that the formula for the nth term of a geometric sequence is $T_n = a r^{n-1}$. We have also seen that the sum of a geometric sequence is given by the formula $S_n = \frac{a(1-r^n)}{1-r}$, and that the difference between a geometric sequence and an arithmetic sequence is that a geometric sequence is found by multiplying the previous term by a fixed, non-zero number, while an arithmetic sequence is found by adding a fixed number.