Problem 90 - Algebra 2

Question

What is the 30 th term of the sequence \(7,16,25,34, \ldots ?\) F. 277 G. 270 H. 268 J. 261

Step-by-Step Solution

Answer

The 30th term of the sequence is 268, so the correct answer is H. 268

Step 1: Identify the first term and the common difference

In this arithmetic sequence, the first term, which we will call \(a\), is 7. The common difference, referred to as \(d\), is the difference between any two successive terms. This is calculated by subtracting the first term from the second term, hence \(d = 16 - 7 = 9.\)

Step 2: Compute the 30th term using the arithmetic sequence formula

The formula for the nth term of an arithmetic sequence is as follows: \(a_n = a + (n-1)*d\). Substituting \(n = 30\), \(a = 7\) and \(d = 9\) into this formula gives: \(a_{30} = 7 + (30-1)*9.\)

Step 3: Simplify to get the final result

By performing the calculations, we find the value of \(a_{30} = 7 + 29*9 = 268.\)

Key Concepts

Common Difference

In arithmetic sequences, the "common difference" is a crucial concept. It defines the difference between any two consecutive terms of the sequence. To find the common difference, you subtract the first term from the second term or any term from its preceding one.
For instance, in the sequence given in the exercise: 7, 16, 25, 34, ..., the common difference is obtained by calculating the difference between the second term and the first term as follows:

• Second term: 16
• First term: 7
• Common Difference, \(d = 16 - 7 = 9\)

This common difference of 9 remains constant throughout the sequence. Having a constant difference signifies you are dealing with an arithmetic sequence.

Nth Term

The "Nth term" of an arithmetic sequence is a formula used to find any term in the sequence without listing all terms up to that point. This calculation allows you to jump straight to the term you need. In formulas, the "Nth term" is often referred to as \(a_n\).
To find the Nth term, you will need the following information:

• First term of the sequence, \(a\)
• Common difference, \(d\)
• Position of the term you want to find, \(n\)

Using this data, apply the arithmetic sequence formula to find your desired term information.

Arithmetic Sequence Formula

The "Arithmetic Sequence Formula" is a mathematical expression used to determine any term in an arithmetic sequence. This formula uses the first term, the common difference, and the position of the term you want to find.
The formula to find the nth term \(a_n\) in an arithmetic sequence is:

• \(a_n = a + (n-1) \times d\)

Where:

• \(a\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference between terms.

For example, to find the 30th term in the sequence from the exercise, substitute \(a = 7\), \(n = 30\), and \(d = 9\) into the formula:

• \(a_{30} = 7 + (30-1) \times 9\)
• Calculate: \(a_{30} = 7 + 29 \times 9\)
• Result: \(a_{30} = 268\)

This step-by-step calculation shows how the formula is applied for efficient results.