Problem 25 - Contemporary Precalculus

Question

The first term \(a_{1}\) and the common difference d of an arithmetic sequence are given. Find the fifth term and the formula for the nth term. $$a_{1}=5, d=2$$

Step-by-Step Solution

Answer

Answer: The 5th term of the arithmetic sequence is 13, and the formula for the nth term of the arithmetic sequence is $$a_n = 5 + (n-1)2$$.

Step 1: Determine the 5th term of the arithmetic sequence

To find the 5th term, we can use the formula for the nth term of the arithmetic sequence and plug in the given values. Here, we have $$a_1=5, d=2$$ and we are trying to find the 5th term, so we set $$n=5$$. Using the formula $$a_n = a_1 + (n-1)d$$, we have: $$a_5 = a_1 + (5-1)d$$ $$a_5 = 5 + (4)2$$ $$a_5 = 5 + 8$$ $$a_5 = 13$$ So, the 5th term of the arithmetic sequence is 13.

Step 2: Determine the formula for the nth term of the arithmetic sequence

Now, we need to find the formula for the nth term of the arithmetic sequence given the values of \(a_1 = 5\) and \(d = 2\). We substitute these values into the general formula $$a_n = a_1 + (n-1)d$$ $$a_n = 5 + (n-1)2$$ This is the formula for the nth term of the arithmetic sequence. To summarize, the 5th term of the arithmetic sequence is 13, and the formula for the nth term of the arithmetic sequence is $$a_n = 5 + (n-1)2$$.

Key Concepts

nth term of an arithmetic sequence

The concept of the nth term of an arithmetic sequence is fundamental to understanding how these sequences work. In an arithmetic sequence, each term is generated by adding a constant amount, known as the common difference, to the previous term. The nth term refers to any specific term that you wish to find within this sequence.

For instance, if you're asked to find the 5th term, as in the exercise example, you're essentially seeking the value of the term that appears in the 5th position of the sequence. The formula \[a_n = a_1 + (n-1)d\], where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) signifies the term number, makes this calculation straightforward.

By plugging the appropriate values into this formula, one can easily deduce any term's value within an arithmetic sequence, ensuring a solid grasp of how sequences progress and change.

common difference

The common difference of an arithmetic sequence is like the steady beat in a song — it's what gives the sequence its rhythm. It's the consistent interval between any two consecutive terms in the sequence, and it can be a positive or negative number.

In our exercise, the common difference is given as \(d=2\). This means that with each new term, 2 is added to the preceding term. The uniformity of this difference is what classifies a sequence as arithmetic. If you ever need to identify the common difference within a sequence, simply subtract one term from the subsequent term.

Importance of the Common Difference

Understanding the common difference is crucial because it allows you to:

• Predict future terms in the sequence
• Find previous terms if needed
• Understand the rate of increase or decrease in a sequence

It’s a foundational element that is pivotal in many mathematical concepts involving sequences.

arithmetic sequence formula

An arithmetic sequence formula is a mathematical recipe that gives you the ability to calculate the value of any term in the sequence. The standard arithmetic sequence formula is \[a_n = a_1 + (n-1)d\], where:

• \(a_n\) is the nth term you want to find
• \(a_1\) is the first term in the sequence
• \(d\) is the common difference
• \(n\) is the term number

This formula is incredibly powerful. It streamlines the process of finding any particular term without calculating all the preceding ones. As demonstrated in the exercise, to find the 5th term of a sequence with a first term of 5 and a common difference of 2, we simply plug these values into the formula to get \(a_5 = 5 + (5-1)\times 2 = 13\).

The arithmetic sequence formula is a key player in solving many mathematical problems, especially those pertaining to sequences and series. Grasping this formula not only bolsters one's algebra skills but also opens the door to understanding further concepts in mathematics.