Problem 18 - Algebra for College Students

Question

If the 8th term of an arithmetic sequence is 37 and the 13 th term is 57 . find the 20 th term. 85

Step-by-Step Solution

Answer

The 20th term is 85.

Step 1: Define the Formula for the n-th Term of an Arithmetic Sequence

The formula for the n-th term of an arithmetic sequence is given by: \( a_n = a_1 + (n - 1) imes d \) where \( a_n \) is the n-th term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 2: Identify Given Terms and Set Up Equations

We are given that the 8th term, \( a_8 = 37 \), and the 13th term, \( a_{13} = 57 \). We can set up the following equations using the formula: \( a_1 + 7d = 37 \) and \( a_1 + 12d = 57 \).

Step 3: Find the Common Difference (d)

Subtract the first equation from the second equation to eliminate \( a_1 \):\[(a_1 + 12d) - (a_1 + 7d) = 57 - 37\]This simplifies to:\[5d = 20\]Solve for \( d \):\[d = \frac{20}{5} = 4\]

Step 4: Determine the First Term (a_1)

Substitute \( d = 4 \) back into the first equation: \( a_1 + 7 \times 4 = 37 \). Simplify to find \( a_1 \):\[a_1 + 28 = 37\]\[a_1 = 37 - 28 = 9\]

Step 5: Calculate the 20th Term

Use the formula for the n-th term again to find \( a_{20} \):\[a_{20} = a_1 + 19d\]Substitute \( a_1 = 9 \) and \( d = 4 \):\[a_{20} = 9 + 19 \times 4\]\[a_{20} = 9 + 76 = 85\]

Key Concepts

n-th term formula

Arithmetic sequences are all about patterns. The n-th term formula helps you find any term within that pattern, provided you know the first term and the common difference. This formula is:

• \( a_n = a_1 + (n - 1) \times d \)
• Where \( a_n \) is the n-th term you're looking for.
• \( a_1 \) is the first term of your sequence, and \( n \) is the position of the term in the sequence.
• \( d \) is the common difference, which is the amount each term increases by.

This formula is essential because it translates the sequence's inherent pattern into a predictable mathematical expression. Once you have the formula and know the values of \( a_1 \) and \( d \), you can calculate any term in the sequence, like the 20th term in this exercise.

common difference

The common difference in an arithmetic sequence is the backbone that binds the sequence together. It's the "step" you take from one term to the next. If every step is equal, then you have an arithmetic sequence.

• To find it, you subtract any term from the subsequent term in the sequence.
• In our exercise, we calculated it as \( d = 4 \) by eliminating the first term from the equations \( a_1 + 7d = 37 \) and \( a_1 + 12d = 57 \).

This subtraction method allows us to determine a crucial piece of information that is universally applicable across the sequence. Knowing \( d \) means not only understanding the sequence's rhythm, but it also facilitates finding any term in the sequence that follows this pattern. Without it, accurately calculating terms would be impossible.

term calculation

Once you have both the first term and the common difference, calculating any term in the sequence becomes straightforward using the n-th term formula. Here's how you do it:

• First, substitute the known values into the formula: \( a_n = a_1 + (n - 1) \times d \).
• For example, to find the 20th term, use \( a_1 = 9 \), \( n = 20 \), and \( d = 4 \).
• The calculation becomes \( a_{20} = 9 + (20 - 1) \times 4 \).
• Simplify it to \( a_{20} = 9 + 76 = 85 \).

This process of term calculation doesn't just give you a number. It gives you an exact position in an orderly progression, reaffirming the predictive power and utility of arithmetic sequences. This method can be applied to any arithmetic sequence, making it an essential tool in your mathematics toolkit.