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Chapter 12: Problem 42

Using factorial notation, write the first five terms of the sequence whose general term is given. \(a_{n}=(3 n) !\)

Short Answer

Expert verified
6, 720, 362880, 479001600, 1307674368000

Step by step solution

01

- Understand the General Term

The given sequence's general term is \(a_n = (3n)!\). This means that for each term, substitute the term's position (n) into the formula and calculate the factorial of the result.
02

- Calculate the First Term

For the first term, substitute \(n = 1\) into the general term: \(a_1 = (3 \times 1)! = 3! = 3 \times 2 \times 1 = 6\).
03

- Calculate the Second Term

For the second term, substitute \(n = 2\) into the general term: \(a_2 = (3 \times 2)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
04

- Calculate the Third Term

For the third term, substitute \(n = 3\) into the general term: \(a_3 = (3 \times 3)! = 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880\).
05

- Calculate the Fourth Term

For the fourth term, substitute \(n = 4\) into the general term: \(a_4 = (3 \times 4)! = 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600\).
06

- Calculate the Fifth Term

For the fifth term, substitute \(n = 5\) into the general term: \(a_5 = (3 \times 5)! = 15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 1307674368000\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sequence calculation
Understanding sequence calculation is essential for solving problems involving patterns and series. A sequence is an ordered list of numbers. The position of each number is called the 'term.' To find terms in a sequence, you need to know its general term 鈥� a formula that explains how to find each term based on its position.
factorial definition
A factorial is a product of all positive integers up to a certain number. It's represented by an exclamation mark (e.g., 5! for 5 factorial, which equals 5 脳 4 脳 3 脳 2 脳 1 = 120). Factorials grow very rapidly and are critical in permutations and combinations in probability and statistics. Factorial of a general term like (3n)! means you calculate the product of all integers from 1 to 3n.
general term substitution
Substituting values into the general term of a sequence is straightforward. Simply replace the variable with the term's position number. For example, if the general term is given as (3n)!, then for the 1st term (substitute n = 1) results in (3 脳 1)! = 3!. For the 2nd term (substitute n = 2), it's (3 脳 2)! = 6!. This substitution method helps in systematically finding each term in the sequence.
mathematical sequences
Mathematical sequences are structured collections of numbers following a specific rule. Arithmetic sequences have constant differences between terms, while geometric sequences have constant ratios. Understanding sequences like (3n)! where each term depends on a factorial helps develop deeper insights in series, progressing from one element to the next systematically.

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