Problem 3 - Precalculus: Mathematics for Calculus

Question

Find the first four terms and the 100 th term of the sequence whose \(n\)th term is given. \(a_{n}=n-3\)

Step-by-Step Solution

Answer

First four terms are -2, -1, 0, 1; the 100th term is 97.

Step 1: Understanding the Sequence Formula

The sequence is defined by the formula for the nth term: \(a_n = n - 3\). This means that each term in the sequence is found by subtracting 3 from its position number \(n\). For example, the first term, \(a_1\), is found by substituting \(n = 1\) into the formula.

Step 2: Calculating the First Term

To find the first term, plug \(n = 1\) into the formula: \(a_1 = 1 - 3 = -2\). Thus, the first term of the sequence is \(-2\).

Step 3: Calculating the Second Term

To find the second term, use \(n = 2\): \(a_2 = 2 - 3 = -1\). Hence, the second term in the sequence is \(-1\).

Step 4: Calculating the Third Term

For the third term, substitute \(n = 3\): \(a_3 = 3 - 3 = 0\). Therefore, the third term of the sequence is \(0\).

Step 5: Calculating the Fourth Term

Next, find the fourth term using \(n = 4\): \(a_4 = 4 - 3 = 1\). The fourth term in the sequence is \(1\).

Step 6: Finding the 100th Term

To calculate the 100th term, substitute \(n = 100\) into the sequence formula: \(a_{100} = 100 - 3 = 97\). This means the 100th term of the sequence is \(97\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a type of number pattern where each term increases or decreases by a constant difference. This difference is called the "common difference."
For example, in the sequence defined by the formula \(a_n = n - 3\), each term is determined by subtracting 3 from the position number \(n\).

• If you look at the first few terms of this sequence, they are \(-2, -1, 0, 1, 2, ...\)
• You can see that the difference between each consecutive term is always 1.

This constant addition (or subtraction) of the same number makes the sequence arithmetic. Such sequences are simple to work with due to their predictable nature.

Term Calculation

Calculating terms in an arithmetic sequence is straightforward. You use the given formula to find each term, one by one.
Take the formula from our sequence for example: \(a_n = n - 3\).
To find each term, simply replace \(n\) with the position number:

• First term: Substitute \(n = 1\), get \(a_1 = 1 - 3 = -2\)
• Second term: Substitute \(n = 2\), get \(a_2 = 2 - 3 = -1\)
• Third term: Substitute \(n = 3\), get \(a_3 = 3 - 3 = 0\)
• Fourth term: Substitute \(n = 4\), get \(a_4 = 4 - 3 = 1\)

Using this step-by-step approach for each position \(n\), you can continue finding as many terms as needed without any confusion.

Nth Term Formula

The nth term formula is a powerful tool to directly determine any term in a sequence without calculating all the previous terms.
For arithmetic sequences, this formula is typically given by \(a_n = a_1 + (n-1) \, d\), where \(a_1\) is the first term and \(d\) is the common difference.
However, if you are given \(a_n\) directly, as in \(a_n = n - 3\), you can plug in any \(n\) to find that specific term immediately:

• If you need the 100th term, substitute \(n = 100\): \(a_{100} = 100 - 3 = 97\)

This allows quick access to any term, especially useful for large values of \(n\). By understanding the nth term formula, you can efficiently explore any position in the sequence.