Problem 27 - Algebra and Trigonometry

Question

Find the indicated term in each arithmetic sequence. $$ \text { 90th term of } 3,-3,-9, \ldots $$

Step-by-Step Solution

Answer

The 90th term is -531.

Step 1: Identify the first term

The first term of the sequence is given as 3. Let the first term be denoted by \(a_1 = 3\).

Step 2: Determine the common difference

The common difference \(d\) is found by subtracting the first term from the second term: \[ d = -3 - 3 = -6 \]

Step 3: Use the formula for the nth term of an arithmetic sequence

The formula for the nth term \(a_n\) of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1) \times d \]

Step 4: Substitute the values into the formula

Substitute the known values into the formula: \[ a_{90} = 3 + (90 - 1) \times (-6) \]

Step 5: Simplify the expression

Calculate the expression inside the parentheses first: \[ 90 - 1 = 89 \] Then multiply by the common difference: \[ 89 \times (-6) = -534 \]Finally, add this product to the first term: \[ a_{90} = 3 - 534 = -531 \]

Key Concepts

common difference

Understanding the concept of the 'common difference' in an arithmetic sequence is key to solving these problems. An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference.
The formula for the common difference is simple: subtract any term by the one before it.

For example: Given the sequence 3, -3, -9, ..., subtract the first term from the second term to get: \( d = -3 - 3 = -6 \)

This constant value, -6, is our common difference. This step is crucial because it helps set up the formulas used in subsequent steps.

nth term formula

The 'nth term formula' in an arithmetic sequence is a powerful tool that allows us to find any term in the sequence without having to list out all preceding terms. The formula is: \[ a_n = a_1 + (n - 1) \times d \]

Where:
• \( a_n \) is the nth term we are looking for.
• \( a_1 \) is the first term.
• \( d \) is the common difference.
• \( n \) is the term position.

Knowing these variables makes it straightforward to plug values into the formula and solve. For example, to find the 90th term of the sequence 3, -3, -9, ...:

\( a_{90} = 3 + (90 - 1) \times (-6) \)

. This formula helps simplify the process by using straightforward arithmetic.

sequence simplification

Sequence simplification involves breaking down the steps to find the nth term without unnecessary complexity. Start by identifying the required elements: the first term and the common difference. Then, use the nth term formula and follow these steps:

• Substitute the known values into the formula: \[ a_{90} = 3 + (90 - 1) \times (-6) \].
• Perform the operations inside the parentheses first: \[ 90 - 1 = 89 \].
• Multiply by the common difference: \[ 89 \times (-6) = -534 \].
• Add this product to the first term: \[ 3 - 534 = -531 \].

By systematically simplifying the expression step by step, you can confidently find any term in the sequence. This method helps eliminate mistakes and ensures a clear path to the solution.