Problem 21 - Algebra and Trigonometry

Question

the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$ a_{n}=2(n+1) ! $$

Step-by-Step Solution

Answer

The first four terms of the sequence are \(4, 12, 48, 240\).

Step 1: Compute the first term of the sequence

To begin, substitute \(n=1\) into the equation. Therefore, \(a_{1}=2(1+1) !=2*2!=4\). So, the first term is 4.

Step 2: Compute the second term of the sequence

Next, substitute \(n=2\) into the equation. Therefore, \(a_{2}=2(2+1) !=2*3!=12\). So, the second term is 12.

Step 3: Compute the third term of the sequence

Now, substitute \(n=3\) into the equation. Therefore, \(a_{3}=2(3+1) !=2*4!=48\). So, the third term is 48.

Step 4: Compute the fourth term of the sequence

Lastly, substitute \(n=4\) into the equation. Therefore, \(a_{4}=2(4+1) !=2*5!=240\). So, the fourth term is 240.

Key Concepts

General Term of a Sequence

In sequences, the general term often represents an expression that can define all terms of the sequence. This expression is a formula where you can substitute an "n" value to find specific terms in the sequence. In the case of our original exercise, the general term is given by the formula:

• \(a_{n} = 2(n+1) !\)

This means that each term can be found by determining the factorial of \((n+1)\), then multiplying it by 2. By understanding the general term, you can easily calculate any term in the sequence, whether it be the first, fifth, or even the hundredth term. The ability to analyze the general term helps in predicting sequences and finding patterns that might not be immediately visible.

Factorials

Factorials are a fundamental concept in mathematics, often denoted with an exclamation mark (!). When you see \((n)!\), it refers to the product of all positive integers up to \(n\). For example:

• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

In the sequence provided in the original exercise, the factorial is calculated for \((n + 1)\), which shifts the typical range by one. Factorials grow quickly, meaning each incremental increase in "n" leads to much larger terms. This aspect makes factorials a powerful tool in arithmetic calculations and helps in understanding the construction of the sequences.

Arithmetic Sequences

Arithmetic sequences are characterized by a constant difference between consecutive terms. Unlike the sequence described in the original exercise, arithmetic sequences don't involve rapidly increasing terms like those derived from factorial calculations.
Instead, in an arithmetic sequence, you start with an initial number and repeatedly add a fixed amount, called the common difference. For instance, the sequence:

• 2, 5, 8, 11, 14,...

Here the common difference is 3. Each term is found by adding 3 to the previous term.
In contrast, sequences with factorials, such as \(a_{n} = 2(n+1)!\), don't fit into the category of arithmetic sequences because their terms don't have a constant difference. Instead, they increase at a varying rate determined by the factorial operation and additional multipliers involved in the general term.