Problem 36 - Beginning and Intermediate Algebra

Question

Two terms of an arithmetic sequence are given in each problem. Find the general term of the sequence, \(a_{n}\), and find the indicated term. $$a_{5}=13, a_{11}=31 ; a_{16}$$

Step-by-Step Solution

Answer

The general term of the arithmetic sequence is \(a_n = 1 + (n - 1)(3)\), and the 16th term, \(a_{16}\), is 46.

Step 1: Find the common difference

The common difference, \(d\), is the difference between any two consecutive terms in an arithmetic sequence. We are given two terms of the sequence, \(a_5 = 13\) and \(a_{11} = 31\). Since we are given a term with a higher index, we can find the common difference by dividing the difference between the two terms by the difference in their indices: $$ d = \frac{a_{11} - a_5}{11 - 5} = \frac{31 - 13}{6} = \frac{18}{6} = 3 $$ The common difference between consecutive terms in the sequence is 3.

Step 2: Find the first term

Now that we have the common difference, we can work backwards to find the first term of the sequence: \(a_1\). To do this, we can use the formula for arithmetic sequences: $$ a_n = a_1 + (n - 1)d $$ Here, we will use the term \(a_5 = 13\). We can plug in the values we have to solve for \(a_1\): $$ 13 = a_1 + (5 - 1)(3) $$ Solve for \(a_1\): $$ a_1 = 13 - (4)(3) = 13 - 12 = 1 $$ The first term of the sequence, \(a_1\), is 1.

Step 3: Find the general term formula

Now that we have both the common difference and the first term, we can write the general formula for the arithmetic sequence: $$ a_n = 1 + (n - 1)(3) $$

Step 4: Find the 16th term

Finally, we use the general term formula to find the 16th term of the sequence, \(a_{16}\): $$ a_{16} = 1 + (16 - 1)(3) = 1 + 15(3) = 1 + 45 = 46 $$ The 16th term of the sequence is 46.

Key Concepts

Understanding the Common Difference in Arithmetic Sequences

The common difference, represented by the symbol \( d \), is a key element in understanding arithmetic sequences. It is the constant amount added (or subtracted) between consecutive terms. This consistency is what makes an arithmetic sequence predictable. Given terms from the sequence, such as \( a_5 = 13 \) and \( a_{11} = 31 \), you can determine \( d \) by finding the difference between the two known terms and then dividing by the number of intervals between them. In our example, the difference between \( a_{11} \) and \( a_5 \) is 18, and there are 6 intervals between these indices. So, the common difference is:

• \( d = \frac{31 - 13}{11 - 5} = \frac{18}{6} = 3 \)

The value 3 tells us that each term increases by 3 from the previous term, allowing us to predict future terms of the sequence with ease.

Discovering the First Term of the Sequence

Once you know the common difference, you can work backward to find the first term of the sequence, \( a_1 \). This is crucial because the first term sets the stage from which the entire sequence unfolds. To calculate \( a_1 \), you can use any known term of the sequence along with the arithmetic sequence formula:\[ a_n = a_1 + (n - 1) \,d \]Say we take \( a_5 = 13 \). With \( d = 3 \) that we've previously discovered, plug these values into the formula:

• \( 13 = a_1 + (5 - 1)(3) \)
• \( 13 = a_1 + 12 \)
• \( a_1 = 13 - 12 = 1 \)

This calculation reveals that the sequence starts at 1, giving us a clear starting point for all future terms.

The General Term Formula of the Sequence

The general term formula of an arithmetic sequence allows you to find any term in the sequence without listing all previous terms. It follows a simple structure that combines both the first term, \( a_1 \), and the common difference. Using the general term formula:\[ a_n = a_1 + (n-1) \,d \]For our specific sequence, knowing \( a_1 = 1 \) and \( d = 3 \), the formula becomes:

• \( a_n = 1 + (n-1)(3) \)

This formula tells us how to compute any term's value, \( a_n \), based on its position \( n \). For example, finding \( a_{16} \):

• \( a_{16} = 1 + (16-1)(3) = 1 + 45 = 46 \)

Thus, the general formula empowers you to quickly determine any term in the sequence, highlighting the organized growth set by its arithmetic nature.