List The First 5 Terms Of The Sequences With The $n^{\text{th}}$ Term:(a) $n^2$(b) $n^2 + 1$(c) $n^2 + 4$(d) $n^2 - 2$(f) $5n^2$(g) $\frac{1}{2}n^2$(h) $\frac{1}{4}n^2$(i)

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Introduction

Sequences are an essential concept in mathematics, and understanding them is crucial for various mathematical operations and problem-solving techniques. A sequence is a list of numbers in a specific order, and each term in the sequence is determined by a formula or rule. In this article, we will explore the first 5 terms of several sequences, each with a different $n^{\text{th}}$ term formula.

Sequence (a): $n^2$

The first sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $n^2$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (a)

$n$	$n^2$
1	1
2	4
3	9
4	16
5	25

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $n^2$ are 1, 4, 9, 16, and 25.

Sequence (b): $n^2 + 1$

The next sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $n^2 + 1$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (b)

$n$	$n^2 + 1$
1	2
2	5
3	10
4	17
5	26

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $n^2 + 1$ are 2, 5, 10, 17, and 26.

Sequence (c): $n^2 + 4$

The next sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $n^2 + 4$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (c)

$n$	$n^2 + 4$
1	5
2	8
3	13
4	20
5	29

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $n^2 + 4$ are 5, 8, 13, 20, and 29.

Sequence (d): $n^2 - 2$

The next sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $n^2 - 2$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (d)

$n$	$n^2 - 2$
1	-1
2	2
3	7
4	14
5	23

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $n^2 - 2$ are -1, 2, 7, 14, and 23.

Sequence (f): $5n^2$

The next sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $5n^2$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (f)

$n$	$5n^2$
1	5
2	20
3	45
4	80
5	125

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $5n^2$ are 5, 20, 45, 80, and 125.

Sequence (g): $\frac{1}{2}n^2$

The next sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $\frac{1}{2}n^2$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (g)

$n$	$\frac{1}{2}n^2$
1	0.5
2	2
3	4.5
4	8
5	12.5

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $\frac{1}{2}n^2$ are 0.5, 2, 4.5, 8, and 12.5.

Sequence (h): $\frac{1}{4}n^2$

The next sequence we will examine is the sequence with the $n^{\text{th}}$ term given by $\frac{1}{4}n^2$. To find the first 5 terms of this sequence, we will substitute $n$ with the values 1, 2, 3, 4, and 5.

Calculating the First 5 Terms of Sequence (h)

$n$	$\frac{1}{4}n^2$
1	0.25
2	1
3	2.25
4	4
5	6.25

As we can see, the first 5 terms of the sequence with the $n^{\text{th}}$ term $\frac{1}{4}n^2$ are 0.25, 1, 2.25, 4, and 6.25.

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Introduction

Sequences are an essential concept in mathematics, and understanding them is crucial for various mathematical operations and problem-solving techniques. In our previous article, we explored the first 5 terms of several sequences with different $n^{\text{th}}$ term formulas. In this article, we will answer some frequently asked questions about sequences to help you better understand this concept.

Q: What is a sequence?

A sequence is a list of numbers in a specific order, and each term in the sequence is determined by a formula or rule.

Q: What is the difference between a sequence and a series?

A sequence is a list of numbers in a specific order, while a series is the sum of the terms of a sequence.

Q: How do I find the $n^{\text{th}}$ term of a sequence?

To find the $n^{\text{th}}$ term of a sequence, you need to substitute the value of $n$ into the formula or rule that determines the sequence.

Q: What is the formula for the $n^{\text{th}}$ term of a sequence?

The formula for the $n^{\text{th}}$ term of a sequence is typically given as $a_n = f(n)$, where $a_n$ is the $n^{\text{th}}$ term and $f(n)$ is the formula or rule that determines the sequence.

Q: How do I calculate the first 5 terms of a sequence?

To calculate the first 5 terms of a sequence, you need to substitute the values of $n$ (1, 2, 3, 4, and 5) into the formula or rule that determines the sequence.

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

An arithmetic sequence is a sequence in which each term is obtained by adding a fixed constant to the previous term, while a geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant.

Q: How do I determine if a sequence is arithmetic or geometric?

To determine if a sequence is arithmetic or geometric, you need to examine the formula or rule that determines the sequence. If the formula involves adding a fixed constant, it is an arithmetic sequence. If the formula involves multiplying the previous term by a fixed constant, it is a geometric sequence.

Q: What is the formula for the sum of the first $n$ terms of a sequence?

The formula for the sum of the first $n$ terms of a sequence is typically given as $S_n = \sum_{i=1}^{n} a_i$, where $S_n$ is the sum and $a_i$ is the $i^{\text{th}}$ term.

Q: How do I calculate the sum of the first $n$ terms of a sequence?

To calculate the sum of the first $n$ terms of a sequence, you need to substitute the values of $a_i$ into the formula and perform the necessary calculations.

Conclusion

Sequences are an essential concept in mathematics, and understanding them is crucial for various mathematical operations and problem-solving techniques. By answering some frequently asked questions about sequences, we hope to have provided you with a better understanding of this concept. If you have any further questions or need help with a specific problem, feel free to ask.

Common Sequences

Here are some common sequences that you may encounter:

• Arithmetic sequence: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
• Geometric sequence: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
• Fibonacci sequence: $a_n = a_{n-1} + a_{n-2}$, where $a_1 = 1$ and $a_2 = 1$.
• Harmonic sequence: $a_n = \frac{1}{n}$, where $n$ is a positive integer.

Sequences in Real-Life Applications

Sequences have numerous real-life applications, including:

• Finance: Sequences are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Sequences are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Sequences are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Sequences are used to develop algorithms and data structures, such as sorting and searching algorithms.

By understanding sequences and their applications, you can develop a deeper appreciation for the mathematical concepts that underlie many real-world phenomena.