How to Find the 5th Term in the Sequence

Oct 13, 2025 by thesweetdesigns.com 113 views

Understanding the General Term of a Sequence

In mathematics, a sequence is a list of numbers in a specific order. The general term of a sequence is a formula that describes each term in the sequence. In this case, the general term of the sequence is given by the formula $$a_n=3n+7$$, where $n$ is the term number.

What is the General Term Formula?

The general term formula is a mathematical expression that describes each term in the sequence. In this case, the formula is $$a_n=3n+7$$. This means that to find the value of the $n$ th term, we need to substitute the value of $n$ into the formula and simplify.

How to Find the 5th Term in the Sequence

To find the 5th term in the sequence, we need to substitute $n=5$ into the general term formula. This will give us the value of the 5th term.

Step-by-Step Solution

Let's break down the solution into step-by-step instructions:

Step 1: Substitute n=5 into the General Term Formula

We need to substitute $n=5$ into the general term formula $$a_n=3n+7$$. This gives us:

a_5=3(5)+7

Step 2: Simplify the Expression

Now, we need to simplify the expression by multiplying 3 and 5, and then adding 7.

a_5=15+7

Step 3: Add 15 and 7

Finally, we need to add 15 and 7 to get the value of the 5th term.

a_5=22

Conclusion

Therefore, the 5th term in the sequence is 22.

Answer Options

Let's compare our answer with the answer options:

A) 25
B) 8
C) 22
D) 17

Our answer, 22, matches option C.

Why is this Important?

Understanding how to find the general term of a sequence is an important skill in mathematics. It allows us to describe and analyze sequences, which is a fundamental concept in algebra and calculus.

Real-World Applications

Sequences and series have many real-world applications, such as:

Modeling population growth
Describing financial transactions
Analyzing data in science and engineering

Tips and Tricks

Here are some tips and tricks to help you find the general term of a sequence:

Make sure to read the problem carefully and understand what is being asked.
Use the general term formula to find the value of each term.
Simplify the expression by following the order of operations (PEMDAS).
Check your answer by plugging it back into the general term formula.

Common Mistakes

Here are some common mistakes to avoid when finding the general term of a sequence:

Not reading the problem carefully and misunderstanding the question.
Not using the general term formula to find the value of each term.
Not simplifying the expression by following the order of operations (PEMDAS).
Not checking the answer by plugging it back into the general term formula.

Conclusion

In conclusion, finding the 5th term in a sequence whose general term is $$a_n=3n+7$$ is a straightforward process that involves substituting the value of $n$ into the general term formula and simplifying the expression. By following the step-by-step instructions and avoiding common mistakes, you can find the value of the 5th term with ease.

Q: What is the general term of a sequence?

A: The general term of a sequence is a formula that describes each term in the sequence. It is a mathematical expression that allows us to find the value of each term in the sequence.

Q: How do I find the general term of a sequence?

A: To find the general term of a sequence, you need to analyze the pattern of the sequence and identify the relationship between each term. You can use algebraic expressions, such as formulas or equations, to describe the pattern.

Q: What is the difference between the general term and the specific term of a sequence?

A: The general term of a sequence is a formula that describes each term in the sequence, while the specific term is a particular value in the sequence. For example, the general term of a sequence might be $$a_n=3n+7$$, while the specific term $$a_5$$ is the value of the 5th term in the sequence.

Q: How do I find the value of a specific term in a sequence?

A: To find the value of a specific term in a sequence, you need to substitute the value of $n$ into the general term formula and simplify the expression. For example, if the general term of a sequence is $$a_n=3n+7$$ and you want to find the value of the 5th term, you would substitute $n=5$ into the formula and simplify.

Q: What are some common mistakes to avoid when finding the general term of a sequence?

A: Some common mistakes to avoid when finding the general term of a sequence include:

Not reading the problem carefully and misunderstanding the question.
Not using the general term formula to find the value of each term.
Not simplifying the expression by following the order of operations (PEMDAS).
Not checking the answer by plugging it back into the general term formula.

Q: How do I know if I have found the correct general term of a sequence?

A: To determine if you have found the correct general term of a sequence, you need to check your answer by plugging it back into the general term formula. If the formula holds true for all terms in the sequence, then you have found the correct general term.

Q: Can I use the general term of a sequence to find the sum of the sequence?

A: Yes, you can use the general term of a sequence to find the sum of the sequence. The sum of a sequence is the sum of all the terms in the sequence. You can use the general term formula to find the value of each term and then add them up to find the sum.

Q: How do I find the sum of a sequence using the general term?

A: To find the sum of a sequence using the general term, you need to use the formula for the sum of a sequence, which is:

S_n = \frac{n}{2}(a_1 + a_n)

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the $n$ th term.

Q: What are some real-world applications of sequences and series?

A: Sequences and series have many real-world applications, such as:

Modeling population growth
Describing financial transactions
Analyzing data in science and engineering
Predicting the behavior of complex systems

Q: How do I know if a sequence is convergent or divergent?

A: To determine if a sequence is convergent or divergent, you need to analyze the behavior of the sequence as $n$ approaches infinity. If the sequence approaches a finite limit, then it is convergent. If the sequence does not approach a finite limit, then it is divergent.

Q: Can I use the general term of a sequence to find the limit of the sequence?

A: Yes, you can use the general term of a sequence to find the limit of the sequence. The limit of a sequence is the value that the sequence approaches as $n$ approaches infinity. You can use the general term formula to find the value of each term and then take the limit as $n$ approaches infinity.

Q: How do I find the limit of a sequence using the general term?

A: To find the limit of a sequence using the general term, you need to use the formula for the limit of a sequence, which is:

\lim_{n\to\infty} a_n = \lim_{n\to\infty} (3n+7)

where $a_n$ is the $n$ th term of the sequence.

Conclusion

In conclusion, finding the general term of a sequence is an important skill in mathematics. It allows us to describe and analyze sequences, which is a fundamental concept in algebra and calculus. By following the step-by-step instructions and avoiding common mistakes, you can find the general term of a sequence with ease.