Chapter 12: Q. 12.15 (page 1169)

Write the first five terms of the sequence whose general term is $a_{n} = \frac{(n - 1)!}{(n + 1)!}$

Short Answer

Expert verified

The first five terms of the sequence with the general term as $a_{n} = \frac{(n - 1)!}{(n + 1)!}$ are $\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20} and \frac{1}{30} .$

Step by step solution

Step 1. Given information

$\begin{array}{l} Thegeneraltermisgivenas \\ a_{n} = \frac{(n - 1)!}{(n + 1)!} \\ Wehavetowritethefirst5termsofthesequence. \end{array}$

Step 2. Calculating the first five terms

We substitute the values 1, 2, 3, 4, 5 into the formula $a_{n} = \frac{(n - 1)!}{(n + 1)!}$

$\begin{array}{l} Weknow \\ If n isapositiveinteger,then \\ \Rightarrow n! = n (n - 1) (n - 2) \dots \\ Now, \\ \begin{matrix} n = 1 & n = 2 & n = 3 & n = 4 & n = 5 \\ \begin{array}{l} \Rightarrow a_{1} = \frac{(1 - 1)!}{(1 + 1)!} \\ = \frac{0!}{2!} \\ = \frac{1}{2 \times 1} \\ = \frac{1}{2} \end{array} & \begin{array}{l} \Rightarrow a_{2} = \frac{(2 - 1)!}{(2 + 1)!} \\ = \frac{1!}{3!} \\ = \frac{1}{3 \times 2 \times 1} \\ = \frac{1}{6} \end{array} & \begin{array}{l} \Rightarrow a_{3} = \frac{(3 - 1)!}{(3 + 1)!} \\ = \frac{2!}{4!} \\ = \frac{2 \times 1}{4 \times 3 \times 2 \times 1} \\ = \frac{1}{12} \end{array} & \begin{array}{l} \Rightarrow a_{4} = \frac{(4 - 1)!}{(4 + 1)!} \\ = \frac{3!}{5!} \\ = \frac{3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \\ = \frac{1}{20} \end{array} & \begin{array}{l} \Rightarrow a_{5} = \frac{(5 - 1)!}{(5 + 1)!} \\ = \frac{4!}{6!} \\ = \frac{4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \\ = \frac{1}{30} \end{array} \end{matrix} \end{array}$

Step 3. Conclusion

From the above step, we can observe that the first five terms of the sequence will be $\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20} and \frac{1}{30} .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Write the first five terms of the sequence whose general term is $a_{n} = \frac{(n - 1)!}{(n + 1)!}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculating the first five terms

Step 3. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Geometry

Discrete Mathematics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Write the first five terms of the sequence whose general term isan=(n-1)!(n+1)!

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculating the first five terms

Step 3. Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Geometry

Discrete Mathematics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Write the first five terms of the sequence whose general term is $a_{n} = \frac{(n - 1)!}{(n + 1)!}$