Question

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? $$ a_{1}=0 ; \quad d=\pi $$

Short answer

The 51st term is \(50\pi\).

Step-by-step solution

Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted as \(d\). The general formula for the nth term of an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \]

Identify Given Values

From the problem, the first term \(a_1\) is given as 0, and the common difference \(d\) is given as \(\pi\). We need to find the 51st term of the sequence.

Plug Values into the Formula

Using the formula for the nth term of an arithmetic sequence, substitute the given values: \(a = 0\), \(d = \pi\), and \(n = 51\). \[a_{51} = a_1 + (51-1)d = 0 + 50\pi\]

Simplify the Expression

Simplify the expression to find the 51st term. Since \(0\) is the first term, the equation becomes: \[a_{51} = 50\pi\]

Key concepts

nth term

In an arithmetic sequence, each term is found by adding a fixed number, known as the common difference, to the previous term. This makes the sequence predictable and easy to work with. The formula for the nth term of an arithmetic sequence is extremely useful. It's written as: \[ a_n = a_1 + (n-1) \times d \]
Let's break down what each part means:

• a_n: The nth term we want to find.
• a_1: The first term in the sequence.
• n: The position of the term in the sequence that we are looking for.
• d: The common difference between the terms.

Knowing how to use this formula allows you to find any term in the sequence quickly.

common difference

The common difference, represented by the symbol \(d\), is what sets an arithmetic sequence apart from other types of sequences. It's the fixed amount that you add (or subtract) to get from one term to the next. In our problem, the common difference \(d\) is given as \(\pi\), which is an interesting twist since \(\pi\) is a well-known irrational number. To clarify:

• Positive Common Difference: If \(d\) is positive, the terms will increase as the sequence progresses.
• Negative Common Difference: If \(d\) is negative, the terms will decrease.

Understanding the common difference helps you grasp how the sequence behaves and evolves.

sequence formula

To solve our specific problem, we use the sequence formula for the nth term:
\[ a_n = a_1 + (n-1) \times d \]
From the problem, we know:

• the first term \(a_1 = 0\)
• the common difference \(d = \pi\)
• and we're looking for the 51st term, so \(n = 51\)

Plugging these values into the formula gives us:
\[ a_{51} = 0 + (51-1) \times \pi \]
Simplifying the expression:
\[ a_{51} = 0 + 50 \pi = 50 \pi \]
So, the 51st term of this arithmetic sequence is \(50 \pi\). Using the formula, you can calculate any term in an arithmetic sequence, as long as you know the first term and the common difference.