/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Find each indicated sum. $$\su... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 30

Find each indicated sum. $$\sum_{i=1}^{5} \frac{(i+2) !}{i !}$$

Short Answer

Expert verified
The sum of the series is 110.

Step by step solution

01

Calculate Factored Terms

We see that the series lies between \(i=1\) to \(i=5\), hence we have 5 terms to calculate. As the terms include factorials, let's try to simplify the equations by canceling out similar factorials. We know that \((i+2)! = (i+2) * (i+1) * i!\), so \(\frac{(i+2) !}{i !}= (i+2) * (i+1)\). Hence, we replace the original terms to get the simplified terms as follows: \n1. For \(i=1\), the term becomes \(3*2\)\n2. For \(i=2\), the term becomes \(4*3\)\n3. For \(i=3\), the term becomes \(5*4\)\n4. For \(i=4\), the term becomes \(6*5\)\n5. For \(i=5\), the term becomes \(7*6\)
02

Compute Each Term of the Sum

Now that we've simplified each term, let's calculate the values for each term: \n1. For \(i=1\), the term equals \(3*2=6\)\n2. For \(i=2\), the term equals \(4*3=12\)\n3. For \(i=3\), the term equals \(5*4=20\)\n4. For \(i=4\), the term equals \(6*5=30\)\n5. For \(i=5\), the term equals \(7*6=42\)
03

Sum all Terms

Finally, we sum up all these terms to get the answer, which is the representation of the original summation expression. Hence, \( \sum_{i=1}^{5} \frac{(i+2) !}{i !}= 6+12+20+30+42=110\).

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