Problem 17 - Algebra and Trigonometry

Question

In Problems 17-24, find the nth term of the arithmetic sequence \(\left\\{a_{n}\right\\}\) whose first term \(a_{1}\) and common difference d are given. What is the 51st term? $$ a_{1}=2 ; \quad d=3 $$

Step-by-Step Solution

Answer

The 51st term is 152.

Step 1: Identify the formula for the nth term

The nth term of an arithmetic sequence can be found using the formula: \[a_{n} = a_{1} + (n - 1) \times d\] where \(a_{1}\) is the first term and \(d\) is the common difference.

Step 2: Substitute the given values into the formula

Given, \(a_{1} = 2\) and \(d = 3\). To find the 51st term, substitute the values into the formula: \[a_{n} = 2 + (51 - 1) \times 3\]

Step 3: Simplify the expression

Calculate the expression inside the parenthesis first: \[a_{n} = 2 + 50 \times 3\]

Step 4: Multiply and add to find the 51st term

Multiply 50 by 3 and add it to 2: \[a_{n} = 2 + 150 = 152\]

Key Concepts

nth term formula

The nth term formula is key to unraveling arithmetic sequences. In any arithmetic sequence, each term increases by a consistent amount known as the common difference. The general formula to find the nth term in such a sequence is: \(a_{n} = a_{1} + (n - 1) \times d\) Here, \(a_{n}\) represents the nth term, \(a_{1}\) is the first term, and \(d\) is the common difference. This formula helps in determining any term in the sequence without listing all terms before it. For example, in our exercise, with \(a_{1} = 2\) and \(d = 3\), substituting these values directly into the formula gives the desired nth term.

common difference

The common difference is a critical parameter in an arithmetic sequence. It tells you how much the sequence increases or decreases with each term. The common difference is denoted by \(d\). In our given problem, the common difference is 3, meaning that each term in the sequence is 3 more than the preceding one. This constant change gives arithmetic sequences their unique structure. Let's consider quick examples:

• If the first term is 2 and the common difference is 5, the sequence will be: 2, 7, 12, 17, ...
• If the first term is 10 and the common difference is -3, the sequence will be: 10, 7, 4, 1, ...

In both cases, the common difference remains consistent.

sequence calculation

Sequence calculation involves applying the nth term formula to obtain specific terms in the sequence. For our exercise, we needed the 51st term with \(a_{1} = 2\) and \(d = 3\). Substituting these values into the nth term formula, we get: \[a_{51} = 2 + (51 - 1) \times 3\] The idea is to calculate step-by-step:

• First, simplify the term inside the parentheses: \(51 - 1 = 50\).
• Then, multiply the result by the common difference: \(50 \times 3 = 150\).
• Finally, add this product to the first term: \(2 + 150 = 152\)

Thus, the 51st term is 152.

simplifying expressions

A significant aspect when working with arithmetic sequences is simplifying expressions. This skill ensures efficiency and precision in finding terms within the sequence. To simplify correctly, follow a methodical approach:

• Identify and isolate the parts of the formula that require computation.
• Perform operations within parentheses first.
• Follow the order of operations, multiplying before adding or subtracting.

In our example, starting with \(a_{51} = 2 + (51 - 1) \times 3\): First simplify inside the parentheses: \(51 - 1 = 50\). Then, multiply 50 by 3 to get 150. Finally, add the first term: \(2 + 150 = 152\). With these steps, you ensure that each part of the sequence calculation is handled accurately, leading to correct results.