Problem 53

Question

In Exercises $53-62,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.) $1,4,7,10, \ldots$

Step-by-Step Solution

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Answer

The $n$ th term of the sequence is given by the expression $3n-2$

1Step 1: Identify the first term $a_1$

In the sequence, the first term $a_1$ is 1.

2Step 2: Find the common difference 'd'

In the given sequence, the difference between consecutive terms is constant. This difference is 3 $= 4 - 1 = 7 - 4 = 10 - 7$. Hence, the common difference 'd' is 3.

3Step 3: Use the general term of an arithmetic sequence

The general form of an arithmetic sequence is $a_n = a_1 + (n-1)*d$. Substituting the values we have: $a_1=1$, 'd'=3, we get $a_n= 1+(n-1)*3 = 1+3n-3 = 3n-2$.

4Step 4: Final Answer

Therefore, the $n$ th term of the sequence is given by the expression $3n-2$

Key Concepts

Common Differencen-th TermGeneral Term of a Sequence

Common Difference

The concept of the common difference is central to understanding arithmetic sequences. In a sequence that is arithmetic, each term after the first is derived by adding a fixed number, called the common difference, to the previous term. This difference remains constant throughout the sequence. For example, consider the sequence provided: 1, 4, 7, 10, ...

The difference between 4 and 1 is 3.
The difference between 7 and 4 is also 3.
Similarly, the difference between 10 and 7 is 3.

This shows that the common difference in our sequence is 3. The presence of a common difference allows for predicting subsequent terms easily by adding this value repeatedly.

n-th Term

The n-th term of a sequence refers to a formula that helps us determine any term in the sequence without listing all the terms. Knowing how to find the n-th term is particularly useful when dealing with large sequences. In arithmetic sequences, the n-th term formula is constructed with the first term and the common difference.

For our sequence,

The first term ($a_1$) is 1.
The common difference ($d$) is 3.

To find the n-th term, we multiply the common difference by the position of the term minus one and then add the first term. This means for the k-th term, the formula is:\[a_n = a_1 + (n-1) \times d\]Using this, the specific term formula adapts to the sequence in question, making predictions of future terms straightforward.

General Term of a Sequence

The general term of an arithmetic sequence can be considered as the most versatile expression to grasp the overall pattern of the sequence. It provides a mathematical expression that relates any term with its position in the sequence.

The generic formula for an arithmetic sequence is given by:\[a_n = a_1 + (n-1) \times d\]This equation allows anyone to compute the value of any term in the sequence if the first term and the common difference are known. In the context of our example:

First term ($a_1$) is 1,
common difference ($d$) is 3,

the general term can be simplified by substitution:\[a_n = 1 + (n-1) \times 3\]Through simplification:\[a_n = 1 + 3n - 3 = 3n - 2\]So, the general term for our sequence can actively generate any term like a mathematical shortcut, avoiding the tedious process of counting each step.

Problem 53

Other exercises in this chapter

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