Problem 21 - Algebra and Trigonometry

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? $$ a_{1}=0 ; \quad d=\frac{1}{2} $$

Step-by-Step Solution

Answer

The 51st term is 25.

Step 1: Understand the Formula for the nth Term of an Arithmetic Sequence

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

Step 2: Substitute the Given Values

Substitute the given values into the formula. We know that \(a_{1} = 0\) and \(d = \frac{1}{2}\). We are looking for the 51st term, so \(n = 51\). Plugging in these values, we get: \[ a_{51} = 0 + (51 - 1) \times \frac{1}{2} \]

Step 3: Simplify the Expression

First calculate the expression inside the parentheses:\[ 51 - 1 = 50 \]Now multiply by the common difference \( \frac{1}{2} \):\[ 50 \times \frac{1}{2} = 25 \]Thus, the 51st term is \[ a_{51} = 25 \]

Key Concepts

nth term

In any arithmetic sequence, the 'nth term' refers to any term in the sequence denoted by the position number 'n'. The formula to find the nth term is very straightforward: \ \ \( a_{n} = a_{1} + (n - 1) \times d \) \ \ Here: \ \ - \(a_{n}\) is the nth term \ - \(a_{1}\) is the first term \ - \(d\) is the common difference \ - \(n\) is the term number \ \ The formula is used to find any term in the sequence. It uses the first term as a starting point and the common difference to calculate how much to add to move between terms.

common difference

The 'common difference' is a key element in an arithmetic sequence. It’s the amount added or subtracted from one term to the next. In the sequence we're dealing with, the common difference (d) is \( \frac{1}{2} \), meaning each term is half more than the previous term. \ \ - If the sequence were \{0, 0.5, 1, 1.5, 2, ...\}, \( \frac{1}{2} \) would be the difference between each term. \ - The formula for the nth term incorporates the common difference directly: \( a_{n} = a_{1} + (n-1) \times d \) \ \ Without the common difference, finding any term would be difficult because you'd have no regular change to count on.

arithmetic progression

An 'arithmetic progression' (or arithmetic sequence) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the 'common difference'. \ \ For our arithmetic sequence where \(a_1 = 0 \) and \(d = \frac{1}{2} \): \ \ - The progression would look like \(0, 0.5, 1, 1.5, 2, ...\) \ - Each number in the sequence is found by adding a constant value (\(\frac{1}{2}\) in this case) to the previous term. \ \ The arithmetic progression can be visually represented as regularly spaced points on a number line or graph.

sequence formula

The 'sequence formula' for an arithmetic sequence allows you to find any term in that sequence directly. The general formula is: \( a_{n} = a_{1} + (n-1) \times d \). This formula eliminates the need to list all the previous terms to find a specific term. \ \ For example, with \( a_{1} = 0 \) and \( d = \frac{1}{2} \), to find the 51st term (\(a_{51}\)): \ \ \( a_{51} = 0 + (51-1) \times \frac{1}{2} \) \ \ \( a_{51} = 0 + 50 \times \frac{1}{2} \) \ \ \( a_{51} = 25 \) \ \ It's a quick way to find any term without counting each one.