Understanding Sequences and Their General Terms

Oct 13, 2025 by magicalceylontours.com 7 views

EXERCISE 9.2: BASIC PRACTICE

Understanding Sequences and Their General Terms

Sequences are an essential concept in mathematics, and understanding how to find their general terms is crucial for solving various mathematical problems. In this exercise, we will focus on finding the first three terms of each sequence given by a general term $$T_n$$. This will help us grasp the concept of sequences and their general terms.

Finding the First Three Terms of Each Sequence

(a) $$T_n = 2n + 1$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = 2n + 1$$.

For $$n = 1$$, $$T_1 = 2(1) + 1 = 3$$
For $$n = 2$$, $$T_2 = 2(2) + 1 = 5$$
For $$n = 3$$, $$T_3 = 2(3) + 1 = 7$$

Therefore, the first three terms of the sequence are $$3, 5, 7$$.

(b) $$T_n = 7 - 3n$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = 7 - 3n$$.

For $$n = 1$$, $$T_1 = 7 - 3(1) = 4$$
For $$n = 2$$, $$T_2 = 7 - 3(2) = 1$$
For $$n = 3$$, $$T_3 = 7 - 3(3) = -2$$

Therefore, the first three terms of the sequence are $$4, 1, -2$$.

(c) $$T_n = 2(n-1)^2$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = 2(n-1)^2$$.

For $$n = 1$$, $$T_1 = 2(1-1)^2 = 0$$
For $$n = 2$$, $$T_2 = 2(2-1)^2 = 2$$
For $$n = 3$$, $$T_3 = 2(3-1)^2 = 8$$

Therefore, the first three terms of the sequence are $$0, 2, 8$$.

(d) $$T_n = \frac{n}{n+2}$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = \frac{n}{n+2}$$.

For $$n = 1$$, $$T_1 = \frac{1}{1+2} = \frac{1}{3}$$
For $$n = 2$$, $$T_2 = \frac{2}{2+2} = \frac{1}{2}$$
For $$n = 3$$, $$T_3 = \frac{3}{3+2} = \frac{3}{5}$$

Therefore, the first three terms of the sequence are $$\frac{1}{3}, \frac{1}{2}, \frac{3}{5}$$.

Conclusion

In this exercise, we have found the first three terms of each sequence given by a general term $$T_n$$. We have seen how to substitute different values of $$n$$ into the general term to find the corresponding terms of the sequence. This exercise has helped us understand the concept of sequences and their general terms, and we are now equipped to find the first three terms of any sequence given by a general term.
EXERCISE 9.2: BASIC PRACTICE

Understanding Sequences and Their General Terms

Finding the First Three Terms of Each Sequence

(a) $$T_n = 2n + 1$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = 2n + 1$$.

For $$n = 1$$, $$T_1 = 2(1) + 1 = 3$$
For $$n = 2$$, $$T_2 = 2(2) + 1 = 5$$
For $$n = 3$$, $$T_3 = 2(3) + 1 = 7$$

Therefore, the first three terms of the sequence are $$3, 5, 7$$.

(b) $$T_n = 7 - 3n$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = 7 - 3n$$.

For $$n = 1$$, $$T_1 = 7 - 3(1) = 4$$
For $$n = 2$$, $$T_2 = 7 - 3(2) = 1$$
For $$n = 3$$, $$T_3 = 7 - 3(3) = -2$$

Therefore, the first three terms of the sequence are $$4, 1, -2$$.

(c) $$T_n = 2(n-1)^2$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = 2(n-1)^2$$.

For $$n = 1$$, $$T_1 = 2(1-1)^2 = 0$$
For $$n = 2$$, $$T_2 = 2(2-1)^2 = 2$$
For $$n = 3$$, $$T_3 = 2(3-1)^2 = 8$$

Therefore, the first three terms of the sequence are $$0, 2, 8$$.

(d) $$T_n = \frac{n}{n+2}$$

To find the first three terms of the sequence, we need to substitute $$n = 1, 2, 3$$ into the general term $$T_n = \frac{n}{n+2}$$.

For $$n = 1$$, $$T_1 = \frac{1}{1+2} = \frac{1}{3}$$
For $$n = 2$$, $$T_2 = \frac{2}{2+2} = \frac{1}{2}$$
For $$n = 3$$, $$T_3 = \frac{3}{3+2} = \frac{3}{5}$$

Therefore, the first three terms of the sequence are $$\frac{1}{3}, \frac{1}{2}, \frac{3}{5}$$.

Q&A

Q: What is a sequence?

A: A sequence is a list of numbers in a specific order, where each number is called a term.

Q: What is a general term?

A: A general term is a formula that describes each term in a sequence.

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

A: To find the first three terms of a sequence, you need to substitute $$n = 1, 2, 3$$ into the general term.

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

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

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

A: If the difference between consecutive terms is constant, the sequence is arithmetic. If the ratio between consecutive terms is constant, the sequence is geometric.

Q: Can a sequence have a negative term?

A: Yes, a sequence can have a negative term.

Q: Can a sequence have a fraction as a term?

A: Yes, a sequence can have a fraction as a term.

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

A: To find the sum of a sequence, you need to add up all the terms in the sequence.

Q: Can a sequence have an infinite number of terms?

A: Yes, a sequence can have an infinite number of terms.

Conclusion

In this exercise, we have found the first three terms of each sequence given by a general term $$T_n$$. We have also answered some common questions about sequences and their general terms. This exercise has helped us understand the concept of sequences and their general terms, and we are now equipped to find the first three terms of any sequence given by a general term.