Problem 19 - Algebra and Trigonometry

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{200}\) when \(a_{1}=-40, d=5\)

Step-by-Step Solution

Answer

The 200th term of the arithmetic sequence is 955.

Step 1: Express the Given Variables

The first term, \(a_{1} = -40\), and the common difference, \(d = 5\). We are asked to find the term \(a_{200}\).

Step 2: Apply the Arithmetic Sequence Formula

We use the formula \(a_{n} = a_{1} + (n - 1) * d\), substituting the given values into it: \(a_{200} = -40 + (200 - 1) * 5\).

Step 3: Solve for \(a_{200}\)

By simplifying the formula we derive the 200th term: \(a_{200} = -40 + 199 * 5 = -40 + 995 = 955\)

Key Concepts

First Term

In an arithmetic sequence, the "first term" is the starting point of the sequence. It is typically denoted by \(a_{1}\). This term sets the foundation for the sequence and determines where your sequence begins. In the context of arithmetic sequences, this first term is crucial because every subsequent term is calculated by adding the common difference to the previous term.

Think of the first term like the beginning step of a long staircase. For example, if you were given that the first term of a sequence is \(-40\), this means that your sequence starts with the number \(-40\), and all future terms will be built upon this initial value.

Common Difference

The "common difference" in an arithmetic sequence is the reason the sequence has its unique, evenly spaced pattern. It is denoted by \(d\). This constant value is what you add to each term to get the next term in the sequence.

For instance, consider a sequence where the common difference is \(5\). This means you start at the first term and add \(5\) to find the next term. If the first term was \(-40\), the second term would be calculated as \(-40 + 5 = -35\). Every term thereafter is reached by continuing to add \(5\). Think of the common difference as the regular step size you use to progress along the number line in your sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula allows you to find any term in the sequence without having to generate all the terms. It is a mathematical shortcut that saves time. The formula is \(a_{n} = a_{1} + (n - 1) \cdot d\), where:

• \(a_{n}\) is the term you want to find (like the 200th term).
• \(a_{1}\) is the first term.
• \(n\) is the position of the term in the sequence (such as 200).
• \(d\) is the common difference.

To use this formula, simply plug in the values you know, solve the arithmetic, and find your desired term. It turns the process from a repetitive addition into a straightforward calculation.

Term Calculation

Term calculation in an arithmetic sequence involves using the arithmetic sequence formula to determine a specific term. With the first term \(-40\) and common difference \(5\), we can find \(a_{200}\) using \(a_{200} = a_{1} + (n - 1) \cdot d\). Plug the known values into the formula:

\[a_{200} = -40 + (200 - 1) \times 5\]

First, calculate \((200 - 1)\) to get \(199\). Next, multiply \(199\) by the common difference \(5\), resulting in \(995\). Finally, add this result to the first term \(-40\):

\(-40 + 995 = 955\).

Thus, the 200th term of the sequence is \(955\). Calculating a term this way ensures accuracy and demonstrates the power of mathematical formulas in simplifying complex operations.