Problem 20 - Algebra and Trigonometry

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{150}\) when \(a_{1}=-60, d=5\)

Step-by-Step Solution

Answer

The 150th term of the given arithmetic sequence is 685.

Step 1: Identify the Given Values

In the given problem, the first term \(a_{1}\) is -60, the common difference \(d\) is 5, and the term to find \(n\) is 150.

Step 2: Apply the Arithmetic Sequence Formula

Apply the arithmetic sequence formula: \(a_{n} = a_{1} + (n - 1) * d\). This simplifies to: \(a_{150} = -60 + (150 - 1) * 5\).

Step 3: Calculate the 150th term

Perform the arithmetic operations to find the value of \(a_{150}\): \(a_{150} = -60 + 149 * 5 = -60 + 745 = 685\).

Key Concepts

Understanding the First Term in Arithmetic Sequences

In any arithmetic sequence, the first term is a foundational component that sets the stage for the entire sequence. It is usually denoted by \(a_1\). This first term determines the starting point of the sequence and so is crucial when calculating future terms.

• The first term is the initial value from which the sequence progresses.
• In our exercise, the first term \(a_1\) is \(-60\), indicating that the sequence begins from \(-60\).
• Without this first term, it would be impossible to discern the exact values of subsequent terms in the sequence.

Remember, every arithmetic sequence relies on knowing its first term to describe the sequence properly.

Deciphering the Common Difference

The common difference, denoted as \(d\), is another important aspect of arithmetic sequences. It represents the consistent interval between consecutive terms in the sequence.

• For our sequence example, the common difference \(d = 5\).
• This means that each term is derived by adding 5 to the previous term, following a consistent increasing pattern.
• The common difference can be positive, negative, or even zero, which affects whether the sequence increases, decreases, or stays the same.

Understanding the common difference helps in predicting and calculating future terms in a sequence without ambiguity.

Applying the Arithmetic Sequence Formula

The arithmetic sequence formula is a powerful tool that helps us calculate any term in a sequence when both the first term and the common difference are known. The formula is:\[a_n = a_1 + (n - 1) \times d\]

• \(a_n\) represents the term you wish to find, where \(n\) is its position in the sequence.
• This formula uses \(a_1\) (the first term) and \(d\) (the common difference) to calculate \(a_n\).
• In our exercise, the 150th term, \(a_{150}\), was calculated by substituting \(-60\) for \(a_1\), \(5\) for \(d\), and \(150\) for \(n\).

This special formula saves a great deal of time by eliminating the need to write out all preceding terms to find any term in the sequence.