Problem 25 - Algebra 2 and Trigonometry

Question

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 10 i, 8 i, 6 i, 4 i, \ldots $$

Step-by-Step Solution

Answer

The ninth term of the sequence is \(-6i\).

Step 1: Identify the sequence type

The sequence provided is: \( 10i, 8i, 6i, 4i, \ldots \) which appears to be an arithmetic sequence as each term decreases by a consistent amount.

Step 2: Determine the common difference

To find the common difference, subtract the second term from the first term: \( 8i - 10i = -2i \). This shows that each term is reduced by \(-2i\).

Step 3: Write the general formula for the arithmetic sequence

The formula for the \(n\)-th term of an arithmetic sequence is: \( a_n = a_1 + (n-1) imes d \), where \(a_1 = 10i\) and \(d = -2i\).

Step 4: Substitute known values into the formula

Substituting the values, the sequence formula becomes \( a_n = 10i + (n-1)(-2i) \).

Step 5: Simplify the expression for the algebraic expression

The expression simplifies to: \( a_n = 10i - 2i(n-1) = 10i - 2in + 2i = (12i - 2in) \).

Step 6: Find the ninth term of the sequence

Substitute \(n = 9\) into the simplified formula: \( a_9 = 12i - 2i imes 9 \).

Step 7: Calculate the ninth term

Perform the calculation: \( a_9 = 12i - 18i = -6i \).

Key Concepts

Common Difference

The common difference is a critical element in understanding arithmetic sequences. It represents the consistent amount by which each term in an arithmetic sequence increases or decreases as you move from one term to the next. In simpler terms, it's the number you add or subtract to get from one term to the next in the sequence.
To find the common difference (d), you simply subtract the first term from the second term.

• For the sequence provided: \(10i, 8i, 6i, 4i, \ldots \), the common difference, \(d\), is the result of \(8i - 10i = -2i\).

This negative common difference signifies a decrease by \(-2i\) in each subsequent term. Understanding this concept allows us to clearly see the pattern within the arithmetic sequence and provides a foundation for creating an algebraic expression that describes the sequence effectively.

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. When working with sequences, it helps us express the relationship between the sequence terms in a general form.
For arithmetic sequences, the algebraic expression is derived using the common difference, initial term, and the position in the sequence (denoted by \(n\)).

• The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \times d\).
• In this particular exercise, substituting the known values yields: \(a_n = 10i + (n-1)(-2i)\).

This general expression is simplified to \(a_n = 12i - 2in\) to express any term in the sequence relative to its position \(n\). This simplified expression is crucial for quickly finding any term in the sequence without recalculating the previous terms.

n-th Term Formula

The \(n\)-th term formula in the context of arithmetic sequences allows for the calculation of any term in the sequence without having to list all preceding terms. This formula makes use of the initial term, the position number, and the common difference.
In the case of this exercise:

• The \(n\)-th term formula derived is \(a_n = 12i - 2in\).
• This formula takes the initial term adjustment \(12i\) and subtracts \(2in\) based on the term’s position \(n\) to find the specific term value.

For example, to find the 9th term, we substitute \(n = 9\) into this formula: \(a_9 = 12i - 2i \times 9\). Perform the simplification to find \(a_9 = -6i\). This approach streamlines solving for specific terms in large or complex sequences efficiently, avoiding errors that might occur in manual calculations.