At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure! Let's complete and solve each series step-by-step.
### 1. Series: [tex]\[12 + 16 + 20 + 24 + \ldots = \sum_{n=1}^{10} (8 + 4n)\][/tex]
We need to determine the series and then find the sum:
1. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 4 \cdot 1 = 8 + 4 = 12 \)[/tex]
2. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 4 \cdot 2 = 8 + 8 = 16 \)[/tex]
3. For [tex]\( n=3 \)[/tex]: [tex]\( 8 + 4 \cdot 3 = 8 + 12 = 20 \)[/tex]
4. For [tex]\( n=4 \)[/tex]: [tex]\( 8 + 4 \cdot 4 = 8 + 16 = 24 \)[/tex]
5. [tex]\(\vdots\)[/tex]
6. For [tex]\( n=10 \)[/tex]: [tex]\( 8 + 4 \cdot 10 = 8 + 40 = 48 \)[/tex]
So, the sequence is: 12, 16, 20, 24, 28, 32, 36, 40, 44, 48.
Now, summing these terms:
[tex]\[ 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48 = 300 \][/tex]
### 2. Series: [tex]\[13 + 17 + 21 + 25 + \ldots + 57 = \sum (5 + 4n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=2 \)[/tex]: [tex]\( 5 + 4 \cdot 2 = 5 + 8 = 13 \)[/tex]
2. For [tex]\( n=3 \)[/tex]: [tex]\( 5 + 4 \cdot 3 = 5 + 12 = 17 \)[/tex]
3. For [tex]\( n=4 \)[/tex]: [tex]\( 5 + 4 \cdot 4 = 5 + 16 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=14 \)[/tex]: [tex]\( 5 + 4 \cdot 14 = 5 + 56 = 61 \)[/tex]
So, the sequence is: 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61.
Now, summing these terms:
[tex]\[ 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49 + 53 + 57 + 61 = 481 \][/tex]
### 3. Series: [tex]\[8 + 11 + 14 + 17 + \ldots + 29 = \sum_{n=2}\][/tex]
To find the series and its sum:
1. For [tex]\( n=0 \)[/tex]: [tex]\( 8 + 3 \cdot 0 = 8 + 0 = 8 \)[/tex]
2. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 3 \cdot 1 = 8 + 3 = 11 \)[/tex]
3. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 3 \cdot 2 = 8 + 6 = 14 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=7 \)[/tex]: [tex]\( 8 + 3 \cdot 7 = 8 + 21 = 29 \)[/tex]
So, the sequence is: 8, 11, 14, 17, 20, 23, 26, 29.
Now, summing these terms:
[tex]\[ 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 = 148 \][/tex]
### 4. Series: [tex]\[12 + 15 + 18 + 21 + \ldots + 39 = \sum (3 + 3n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=4 \)[/tex]: [tex]\( 3 + 3 \cdot 4 = 3 + 12 = 15 \)[/tex]
2. For [tex]\( n=5 \)[/tex]: [tex]\( 3 + 3 \cdot 5 = 3 + 15 = 18 \)[/tex]
3. For [tex]\( n=6 \)[/tex]: [tex]\( 3 + 3 \cdot 6 = 3 + 18 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=11 \)[/tex]: [tex]\( 3 + 3 \cdot 11 = 3 + 33 = 36 \)[/tex]
So, the sequence is: 15, 18, 21, 24, 27, 30, 33, 36.
Now, summing these terms:
[tex]\[ 15 + 18 + 21 + 24 + 27 + 30 + 33 + 36 = 204 \][/tex]
Therefore, the completed series and sums are as follows:
1. [tex]\(12, 16, 20, 24, 28, 32, 36, 40, 44, 48 = \sum (8 + 4n) = 300\)[/tex]
2. [tex]\(13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61 = \sum (5 + 4n) = 481\)[/tex]
3. [tex]\(8, 11, 14, 17, 20, 23, 26, 29 = \sum_{n=2} = 148\)[/tex]
4. [tex]\(15, 18, 21, 24, 27, 30, 33, 36 = \sum (3 + 3n) = 204\)[/tex]
### 1. Series: [tex]\[12 + 16 + 20 + 24 + \ldots = \sum_{n=1}^{10} (8 + 4n)\][/tex]
We need to determine the series and then find the sum:
1. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 4 \cdot 1 = 8 + 4 = 12 \)[/tex]
2. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 4 \cdot 2 = 8 + 8 = 16 \)[/tex]
3. For [tex]\( n=3 \)[/tex]: [tex]\( 8 + 4 \cdot 3 = 8 + 12 = 20 \)[/tex]
4. For [tex]\( n=4 \)[/tex]: [tex]\( 8 + 4 \cdot 4 = 8 + 16 = 24 \)[/tex]
5. [tex]\(\vdots\)[/tex]
6. For [tex]\( n=10 \)[/tex]: [tex]\( 8 + 4 \cdot 10 = 8 + 40 = 48 \)[/tex]
So, the sequence is: 12, 16, 20, 24, 28, 32, 36, 40, 44, 48.
Now, summing these terms:
[tex]\[ 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 + 44 + 48 = 300 \][/tex]
### 2. Series: [tex]\[13 + 17 + 21 + 25 + \ldots + 57 = \sum (5 + 4n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=2 \)[/tex]: [tex]\( 5 + 4 \cdot 2 = 5 + 8 = 13 \)[/tex]
2. For [tex]\( n=3 \)[/tex]: [tex]\( 5 + 4 \cdot 3 = 5 + 12 = 17 \)[/tex]
3. For [tex]\( n=4 \)[/tex]: [tex]\( 5 + 4 \cdot 4 = 5 + 16 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=14 \)[/tex]: [tex]\( 5 + 4 \cdot 14 = 5 + 56 = 61 \)[/tex]
So, the sequence is: 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61.
Now, summing these terms:
[tex]\[ 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49 + 53 + 57 + 61 = 481 \][/tex]
### 3. Series: [tex]\[8 + 11 + 14 + 17 + \ldots + 29 = \sum_{n=2}\][/tex]
To find the series and its sum:
1. For [tex]\( n=0 \)[/tex]: [tex]\( 8 + 3 \cdot 0 = 8 + 0 = 8 \)[/tex]
2. For [tex]\( n=1 \)[/tex]: [tex]\( 8 + 3 \cdot 1 = 8 + 3 = 11 \)[/tex]
3. For [tex]\( n=2 \)[/tex]: [tex]\( 8 + 3 \cdot 2 = 8 + 6 = 14 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=7 \)[/tex]: [tex]\( 8 + 3 \cdot 7 = 8 + 21 = 29 \)[/tex]
So, the sequence is: 8, 11, 14, 17, 20, 23, 26, 29.
Now, summing these terms:
[tex]\[ 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 = 148 \][/tex]
### 4. Series: [tex]\[12 + 15 + 18 + 21 + \ldots + 39 = \sum (3 + 3n)\][/tex]
To find the series and its sum:
1. For [tex]\( n=4 \)[/tex]: [tex]\( 3 + 3 \cdot 4 = 3 + 12 = 15 \)[/tex]
2. For [tex]\( n=5 \)[/tex]: [tex]\( 3 + 3 \cdot 5 = 3 + 15 = 18 \)[/tex]
3. For [tex]\( n=6 \)[/tex]: [tex]\( 3 + 3 \cdot 6 = 3 + 18 = 21 \)[/tex]
4. [tex]\(\vdots\)[/tex]
5. For [tex]\( n=11 \)[/tex]: [tex]\( 3 + 3 \cdot 11 = 3 + 33 = 36 \)[/tex]
So, the sequence is: 15, 18, 21, 24, 27, 30, 33, 36.
Now, summing these terms:
[tex]\[ 15 + 18 + 21 + 24 + 27 + 30 + 33 + 36 = 204 \][/tex]
Therefore, the completed series and sums are as follows:
1. [tex]\(12, 16, 20, 24, 28, 32, 36, 40, 44, 48 = \sum (8 + 4n) = 300\)[/tex]
2. [tex]\(13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61 = \sum (5 + 4n) = 481\)[/tex]
3. [tex]\(8, 11, 14, 17, 20, 23, 26, 29 = \sum_{n=2} = 148\)[/tex]
4. [tex]\(15, 18, 21, 24, 27, 30, 33, 36 = \sum (3 + 3n) = 204\)[/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
