Problem 6 - Algebra and Trigonometry

Question

Multiple Choice If \(a_{n}=-2 n+7\) is the \(n\) th term of an arithmetic sequence, the first term is _____. (a) -2 (b) 0 (c) 5 (d) 7

Step-by-Step Solution

Answer

The first term is (c) 5.

Step 1: Understand the Formula

The formula given is for the nth term of an arithmetic sequence: \(a_n = -2n + 7\). To find the first term, substitute \(n = 1\) into the formula.

Step 2: Substitute n with 1

Substitute \(n = 1\) into the nth term formula to find the first term. \[a_1 = -2(1) + 7\]

Step 3: Perform the Calculation

Perform the arithmetic operations: \[-2(1) = -2\]\[-2 + 7 = 5\]Therefore, \(a_1 = 5\).

Step 4: Select the Correct Option

From the given multiple-choice options, match the calculated first term (5) to the correct letter. The correct option is (c) 5.

Key Concepts

nth term

In any arithmetic sequence, the nth term formula is key to understanding the sequence's structure. The provided exercise gives the nth term as: \(a_{n} = -2n + 7\). This formula allows us to calculate the value of any term in the sequence by simply plugging in the term's position (n). The nth term formula is particularly useful because: * It reveals the pattern of the sequence * It provides a straightforward method to find any term without having to list all previous terms * It helps identify other sequence characteristics To effectively use this formula, it's essential to understand each component: the coefficient of n (-2 in this case) defines the sequence's common difference, and the constant term (7) affects the initial values.

first term

Finding the first term of an arithmetic sequence is simple with the nth term formula. For the given exercise, we substitute \(n = 1\) into \(-2n + 7\). Steps: 1. Identify the given nth term formula: \(a_{n} = -2n + 7\) 2. Set \(n \) to 1 to find the first term: \(-2(1) + 7\) 3. Calculate: \(-2(1) = -2\), then add 7 (\[-2 + 7 = 5\]). The first term \(a_{1}\) is 5. This approach can find any specific term by adjusting n. Understanding the first term is crucial as it serves as the starting point for the sequence.

substitution

Substitution is a fundamental mathematical concept used to simplify and solve equations. In the context of sequences, it helps find specific terms, like in the given exercise. When you see a formula, such as the nth term formula \(a_{n} = -2n + 7\), substitution involves: * Identifying the variable to replace (here, n) * Replacing the variable with a specific value (substitute \(n \) with 1 for the first term) * Solving the resulting expression to find your answer In the example: 1. \(n = 1\) 2. Substitute into the formula: \(-2(1) + 7\) 3. Perform calculations to find the answer. Substituting is straightforward but immensely powerful in simplifying complex formulas and finding precise values.