Problem 49 - Precalculus: Mathematics for Calculus

Question

Find the indicated term of the arithmetic sequence with the given description. The first term is \(25,\) and the common difference is \(18 .\) Which term of the sequence is \(601 ?\)

Step-by-Step Solution

Answer

The 33rd term of the sequence is 601.

Step 1: Understand the Problem

We are given the first term of an arithmetic sequence as \( a_1 = 25 \) and the common difference \( d = 18 \). We need to find which term \( a_n \) equals \( 601 \).

Step 2: Use the Formula for the nth Term

In an arithmetic sequence, the nth term can be calculated as \( a_n = a_1 + (n-1) \cdot d \). We need to substitute the known values to solve for \( n \).

Step 3: Substitute Known Values

Substitute \( a_n = 601 \), \( a_1 = 25 \), and \( d = 18 \) into the formula: \[601 = 25 + (n-1) \cdot 18\].

Step 4: Solve for n

First, simplify the equation: \[601 = 25 + 18n - 18\]Then, simplify further:\[601 = 7 + 18n\]Subtract 7 from both sides:\[594 = 18n\].Finally, divide both sides by 18 to isolate \( n \):\[n = \frac{594}{18} = 33\].

Step 5: Verification

Re-evaluate the steps to ensure there are no mistakes. Substitute \( n = 33 \) back into the formula to check if it calculates back to 601:\[a_{33} = 25 + 32 \cdot 18 = 601\], confirming the correct solution.

Key Concepts

nth term

The term \(a_n\) in an arithmetic sequence is crucial to identifying any member of the sequence. Imagine you have a list of numbers generated by regularly adding or subtracting the same number. This number is known as the 'common difference', and it consistently transforms each term to the next. To precisely describe any term in the sequence without listing all the previous numbers, you use the formula for the **nth term**.
To find the nth term, the formula is:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_1\) is the first term, and \(d\) is the common difference. This formula helps you directly calculate any term in the sequence by leveraging the position \(n\) of that term. This way, you don't have to add that difference over and over again manually.

common difference

The 'common difference' is a defining feature of an arithmetic sequence. It is this value that gives the sequence its regular and predictive pattern. Once you know the common difference, you can continue a sequence indefinitely in either direction.
The common difference can be calculated by subtracting any term in the sequence from the succeeding one. If the difference is constant throughout, then the sequence is indeed arithmetic.
For example:

• Consider the sequence 3, 7, 11, 15. The common difference \(d\) is 4.
• This is found by subtracting 3 from 7, 7 from 11, etc.

The concept of common difference not only helps in finding out any term out of sequence but also plays a crucial role in determining the \