Problem 19 - 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? $$ a_{1}=8 ; \quad d=-7 $$

Step-by-Step Solution

Answer

The 51st term is -342.

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 \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number we want to find.

Step 2 - Identify Given Values

From the problem, the first term \( a_1 \) is 8 and the common difference \( d \) is -7. We need to find the 51st term, so \( n \) is 51.

Step 3 - Substitute the Values into the Formula

Substitute the known values into the arithmetic sequence formula: \[ a_{51} = 8 + (51-1)(-7) \]

Step 4 - Simplify the Expression

Simplify inside the parentheses first: \[ a_{51} = 8 + (50)(-7) \]Multiply 50 by -7: \[ a_{51} = 8 - 350 \]Finally, perform the subtraction: \[ a_{51} = -342 \]

Key Concepts

nth term

In any arithmetic sequence, we can find any term if we know the first term and the common difference. The formula used is: yArithmeticy sequences have a specific addition each step, defined by the common difference. The term we want is called the 'nth term'. We use the formula: \[ a_n = a_1 + (n-1)d \] here:

• \( a_n \): the nth term
• \( a_1 \): the first term
• \( n \): the term number
• \( d \): the common difference, the fixed number added or subtracted each step

For example, in the provided problem, the first term \( a_1 \) is 8 and the common difference \( d \) is -7. To find the 51st term (\( n = 51 \)), we substitute the values into the formula: \[ a_{51} = 8 + (51-1)(-7) \] After simplifying, we get: \[ a_{51} = 8 + (50)(-7) \] \[ a_{51} = 8 - 350 \] \[ a_{51} = -342 \] So, the 51st term is -342.

common difference

The common difference in an arithmetic sequence is a key component. It is the uniform amount added (or subtracted) to move from one term to the next. If an arithmetic sequence is represented as \( a_1, a_2, a_3, ... \), where \( a_1 \) is the first term, then: \[ d = a_2 - a_1 \] This difference \( d \) remains constant throughout the sequence. In the problem provided, the common difference \( d \) is -7. This means each term is 7 less than the previous one. To illustrate:

• If the first term \( a_1 \) is 8, then the second term \( a_2 \) is 8 - 7 = 1
• The third term \( a_3 \) is 1 - 7 = -6
• And so forth. The sequence continues by subtracting 7 each step

Understanding the common difference helps in identifying patterns and calculating specific terms using the nth term formula.

arithmetic series formula

While an arithmetic sequence involves the terms individually, an arithmetic series is the sum of these terms. The formula to find the sum of the first n terms of an arithmetic series is: \[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \] where:

• \( S_n \) is the sum of the first n terms
• \( n \) is the number of terms
• \( a_1 \) is the first term
• \( d \) is the common difference

For example, if we were to find the sum of the first 51 terms in our sequence with the first term \( a_1 = 8 \) and common difference \( d = -7 \), the formula would be used as follows: First, calculate: \[ 2a_1 + (n-1)d = 2(8) + (51-1)(-7) = 16 + 50(-7) = 16 - 350 = -334 \] Then the sum: \[ S_{51} = \frac{51}{2} (-334) = 25.5 (-334) = -8517 \] This sum helps in larger calculations and gives insights into the overall progression of the sequence.