Problem 24
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? $$ a_{1}=0 ; \quad d=\pi $$
Step-by-Step Solution
Verified Answer
The 51st term is \(50\pi\).
1Step 1: 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: \[ a_n = a_1 + (n-1)d \]
2Step 2: 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.
3Step 3: 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 \(n = 51\). \[a_{51} = a_1 + (51-1)d = 0 + 50\pi\]
4Step 4: 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 termcommon differencesequence formula
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:
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.
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.
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:
\[ a_{51} = 0 + (51-1) \times \pi \]
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.
\[ a_n = a_1 + (n-1) \times d \]
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 \(n = 51\)
\[ a_{51} = 0 + (51-1) \times \pi \]
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.
Other exercises in this chapter
Problem 24
Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=1 ; \quad r=-\frac{1}{3} $$
View solution Problem 24
List the first five terms of each sequence. \(\left\\{a_{n}\right\\}=\left\\{\frac{3^{n}}{n}\right\\}\)
View solution Problem 25
Prove each statement. $$ \begin{aligned} &a-b \text { is a factor of } a^{n}-b^{n}\\\ &\text { [Hint: } \left.a^{k+1}-b^{k+1}=a\left(a^{k}-b^{k}\right)+b^{k}(a-
View solution Problem 25
Expand each expression using the Binomial Theorem. $$ (\sqrt{x}+\sqrt{2})^{6} $$
View solution