Find the indicated (shown) sum: 5i!1 (і — 1)!WI

We have to find the sum: ∑i=15i!i-1! Here ∑ means addition of all the terms after putting i=1 to i=5…

Answered: 5 i! 1 (і — 1)! WI

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 1E

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Question

Find the indicated (shown) sum:

Expert Solution

Step 1

We have to find the sum:

$\sum_{i = 1}^{5} \frac{i!}{(i - 1)!}$

Here $\sum_{}^{}$ means addition of all the terms after putting $i = 1$ to $i = 5$ into the expression

so putting values of $i$ from $1$ to $5$ in a sequence, we get

$\begin{array}{rcl} \sum_{i = 1}^{5} \frac{i!}{(i - 1)!} & = & \frac{1!}{(1 - 1)!} + \frac{2!}{(2 - 1)!} + \frac{3!}{(3 - 1)!} + \frac{4!}{(4 - 1)!} + \frac{5!}{(5 - 1)!} \\ = & \frac{1!}{(0)!} + \frac{2!}{(1)!} + \frac{3!}{(2)!} + \frac{4!}{(3)!} + \frac{5!}{(4)!} \end{array}$

Here $!$ denotes the factorial

if we have $n!$ then its value will be written in following ways,

$n! = n (n - 1) (n - 2) \times . . . . \times 3 \times 2 \times 1$

or,

$n! = 1 \times 2 \times 3 \times . . . . \times (n - 2) (n - 1) n$

So finding values of the terms,

$\begin{array}{rcl} 0! & = & 1 \\ 1! & = & 1 \\ 2! & = & 2 \times 1 \\ 3! & = & 3 \times 2 \times 1 \\ 4! & = & 4 \times 3 \times 2 \times 1 \\ 5! & = & 5 \times 4 \times 3 \times 2 \times 1 \end{array}$

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