Problem 20 - 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? $$ a_{1}=6 ; \quad d=-2 $$

Step-by-Step Solution

Answer

The 51st term is -94.

Step 1: Understand the arithmetic sequence formula

The formula to find the nth term of an arithmetic sequence is given by \[ a_n = a_1 + (n - 1)d \] 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 known values

Substitute the given values into the arithmetic sequence formula. We know that \( a_1 = 6 \), \( d = -2 \), and we want to find the 51st term (\( n = 51 \)).Using the formula, we substitute the values: \[ a_{51} = 6 + (51 - 1)(-2) \]

Step 3: Simplify the expression

First, simplify the term inside the parentheses: \[ a_{51} = 6 + (50)(-2) \].Next, calculate the product:\[ 50 \times -2 = -100 \].

Step 4: Complete the calculation

Finally, add the results from the previous steps:\[ a_{51} = 6 + (-100) \].Simplify the expression to find the 51st term:\[ a_{51} = 6 - 100 = -94 \].

Key Concepts

nth term formula

To find any term in an arithmetic sequence, we use a special formula. This formula helps us locate a specific term without listing all the terms. The formula is: \ \[ a_n = a_1 + (n - 1)d \] \ Here: \

• \( a_n \): the nth term we are looking for
• \( a_1 \): the first term of the sequence
• \( d \): the common difference between the terms
• \( n \): the term number we need

Using this formula, we can calculate any term if we know the first term and the common difference.

common difference

The common difference in an arithmetic sequence tells us how much each term increases or decreases from the previous one. It is represented by \( d \). \

• If \( d \) is positive, the sequence increases.
• If \( d \) is negative, the sequence decreases.

\ For example, if we know the first term is 6 and the common difference is \( d = -2 \), every next term is 2 units less than the previous term. So, the first few terms will look like 6, 4, 2, 0, -2, ... and so on.

sequence calculation

Let’s use an example to understand how to calculate a specific term in an arithmetic sequence. We will find the 51st term for the sequence with the first term \( a_1 = 6 \) and the common difference \( d = -2 \). \

• Start with the nth term formula: \ \[ a_n = a_1 + (n - 1)d \]
• Substitute the given values: \ \[ a_{51} = 6 + (51 - 1)(-2) \]
• Simplify inside the parenthesis: \ \[ 51 - 1 = 50 \], so \ \[ a_{51} = 6 + 50(-2) \]
• Calculate the product: \ \[ 50 \times -2 = -100 \], so \ \[ a_{51} = 6 + (-100) \]
• Combine the terms: \ \[ 6 - 100 = -94 \]

So, the 51st term of the sequence is \( -94 \). This approach helps us quickly find any term in the sequence without listing all previous terms.