Chapter 12: Q. 12.16 (page 1169)

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

Short Answer

Expert verified

The first five terms of the sequence with general term $a_{n} = \frac{n!}{(n + 1)!}$ are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} and \frac{1}{6} .$

Step by step solution

Step 1. Given information

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

Step 2. Calculating the first five terms

We substitute the values 1, 2, 3, 4, 5 into the formula for the general term.

$\begin{array}{l} Weknow \\ If n isapositiveinteger,then \\ \Rightarrow n! = n (n - 1) (n - 2) \dots \end{array}$

Now,

$\begin{matrix} n = 1 & n = 2 & n = 3 & n = 4 & n = 5 \\ \begin{array}{l} \Rightarrow a_{1} = \frac{1!}{(1 + 1)!} \\ = \frac{1!}{2!} \\ = \frac{1}{2 \times 1} \\ = \frac{1}{2} \end{array} & \begin{array}{l} \Rightarrow a_{2} = \frac{2!}{(2 + 1)!} \\ = \frac{2 \times 1!}{3!} \\ = \frac{2}{3 \times 2 \times 1} \\ = \frac{1}{3} \end{array} & \begin{array}{l} \Rightarrow a_{3} = \frac{3!}{(3 + 1)!} \\ = \frac{3!}{4!} \\ = \frac{3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \\ = \frac{1}{4} \end{array} & \begin{array}{l} \Rightarrow a_{4} = \frac{4!}{(4 + 1)!} \\ = \frac{4!}{5!} \\ = \frac{4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \\ = \frac{1}{5} \end{array} & \begin{array}{l} \Rightarrow a_{5} = \frac{5!}{(5 + 1)!} \\ = \frac{5!}{6!} \\ = \frac{5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \\ = \frac{1}{6} \end{array} \end{matrix}$

Step 3. Conclusion

From the above step, we can observe that the first five terms of the sequence will be $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} and \frac{1}{6} .$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculating the first five terms

Step 3. Conclusion

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Write the first five terms of the sequence whose general term is an=n!(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

Decision Maths

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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