Connect with knowledgeable individuals and get your questions answered on IDNLearn.com. Ask your questions and receive detailed and reliable answers from our experienced and knowledgeable community members.
Sagot :
Certainly! Let's answer each part of Exercise 8.2 in detail:
### 1. (a) Define sequence with a suitable example.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. For example, a sequence could be [tex]\(1, 2, 3, 4, 5, \ldots\)[/tex]. Here, each number follows the previous one in a specific order and the order in which they appear is important.
### 1. (b) What is the difference between sequence and series?
A sequence is a list of numbers arranged in a specific order. For example, the sequence [tex]\(1, 2, 3, 4, 5\)[/tex].
A series, on the other hand, refers to the sum of the elements of a sequence. For example, the series [tex]\(1 + 2 + 3 + 4 + 5\)[/tex].
### 2. Identify sequence and series from the given expressions:
(a) [tex]\(3, 6, 9, 12, 15, \ldots\)[/tex]
- This is a sequence.
- Sequence: [tex]\([3, 6, 9, 12, 15]\)[/tex].
(b) [tex]\(2 + 4 + 6 + 8 + 10 + \ldots\)[/tex]
- This is a series.
- Series: [tex]\([2, 4, 6, 8, 10]\)[/tex].
(c) [tex]\(4, 2, 1, \frac{1}{2}, \frac{1}{4}\)[/tex]
- This is a sequence.
- Sequence: [tex]\([4, 2, 1, \frac{1}{2}, \frac{1}{4}]\)[/tex].
(d) [tex]\(\sum_{n=1}^3(3 n+1)\)[/tex]
- This is a series formed by the sum of an expression evaluated over a range.
- Series: [tex]\([\ (3 \cdot 1 + 1), (3 \cdot 2 + 1), (3 \cdot 3 + 1)] = [4, 7, 10]\)[/tex].
(e) [tex]\(1 + 4 + 9 + 16 + \ldots\)[/tex]
- This is a series.
- Series: [tex]\([1, 4, 9, 16]\)[/tex].
(f) [tex]\(\frac{1}{5}, \frac{3}{8}, \frac{5}{11}, \frac{7}{14}\)[/tex]
- This is a sequence.
- Sequence: [tex]\([0.2, 0.375, 0.45454545454545453, 0.5]\)[/tex].
### 3. Find the value of:
(a) [tex]\(\sum_{n=0}^4(2 n-1)\)[/tex]
- Sum: 15.
(b) [tex]\(\sum_{n=2}^6(3 n+2)\)[/tex]
- Sum: 70.
(c) [tex]\(\sum_{n=1}^5(n^2+1)\)[/tex]
- Sum: 60.
(d) [tex]\(\sum_{n=1}^3(n^2+2 n+1)\)[/tex]
- Sum: 29.
(e) [tex]\(\sum_{n=1}^{10} 5n\)[/tex]
- Sum: 275.
(f) [tex]\(\sum_{n=5}^{10} n^2\)[/tex]
- Sum: 355.
(g) [tex]\(\sum_{n=3}^8(n^2-2)\)[/tex]
- Sum: 187.
(h) [tex]\(\sum_{n=1}^5 \frac{2n+1}{n}\)[/tex]
- Sum: 12.283333333333333.
(i) [tex]\(\sum_{n=0}^4 \frac{n}{n+1}\)[/tex]
- Sum: 2.716666666666667.
### 4. Write the given series using symbol [tex]\(\sum\)[/tex]:
(a) [tex]\(5 + 7 + 9 + 11 + \ldots + 21\)[/tex]
- Series: [tex]\(\sum_{n=0}^{8}(5 + 2n)\)[/tex].
(b) [tex]\(2 + 4 + 6 + 8 + 10 + 12\)[/tex]
- Series: [tex]\(\sum_{n=1}^{6}(2n)\)[/tex].
(c) [tex]\(30 + 25 + 20 + \ldots + 5\)[/tex]
- Series: [tex]\(\sum_{n=0}^{5}(30 - 5n)\)[/tex].
(d) [tex]\(1 + 5 + 9 + 13 + 17 + 21\)[/tex]
- Series: [tex]\(\sum_{n=0}^{5}(1 + 4n)\)[/tex].
(e) [tex]\(1 + 4 + 9 + 16\)[/tex]
- Series: [tex]\(\sum_{n=1}^{4}(n^2)\)[/tex].
(f) [tex]\(a + ab^1 + ab^2 + ab^3 + ab^4\)[/tex]
- Series: [tex]\(\sum_{n=0}^{4}(ab^n)\)[/tex].
### 1. (a) Define sequence with a suitable example.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. For example, a sequence could be [tex]\(1, 2, 3, 4, 5, \ldots\)[/tex]. Here, each number follows the previous one in a specific order and the order in which they appear is important.
### 1. (b) What is the difference between sequence and series?
A sequence is a list of numbers arranged in a specific order. For example, the sequence [tex]\(1, 2, 3, 4, 5\)[/tex].
A series, on the other hand, refers to the sum of the elements of a sequence. For example, the series [tex]\(1 + 2 + 3 + 4 + 5\)[/tex].
### 2. Identify sequence and series from the given expressions:
(a) [tex]\(3, 6, 9, 12, 15, \ldots\)[/tex]
- This is a sequence.
- Sequence: [tex]\([3, 6, 9, 12, 15]\)[/tex].
(b) [tex]\(2 + 4 + 6 + 8 + 10 + \ldots\)[/tex]
- This is a series.
- Series: [tex]\([2, 4, 6, 8, 10]\)[/tex].
(c) [tex]\(4, 2, 1, \frac{1}{2}, \frac{1}{4}\)[/tex]
- This is a sequence.
- Sequence: [tex]\([4, 2, 1, \frac{1}{2}, \frac{1}{4}]\)[/tex].
(d) [tex]\(\sum_{n=1}^3(3 n+1)\)[/tex]
- This is a series formed by the sum of an expression evaluated over a range.
- Series: [tex]\([\ (3 \cdot 1 + 1), (3 \cdot 2 + 1), (3 \cdot 3 + 1)] = [4, 7, 10]\)[/tex].
(e) [tex]\(1 + 4 + 9 + 16 + \ldots\)[/tex]
- This is a series.
- Series: [tex]\([1, 4, 9, 16]\)[/tex].
(f) [tex]\(\frac{1}{5}, \frac{3}{8}, \frac{5}{11}, \frac{7}{14}\)[/tex]
- This is a sequence.
- Sequence: [tex]\([0.2, 0.375, 0.45454545454545453, 0.5]\)[/tex].
### 3. Find the value of:
(a) [tex]\(\sum_{n=0}^4(2 n-1)\)[/tex]
- Sum: 15.
(b) [tex]\(\sum_{n=2}^6(3 n+2)\)[/tex]
- Sum: 70.
(c) [tex]\(\sum_{n=1}^5(n^2+1)\)[/tex]
- Sum: 60.
(d) [tex]\(\sum_{n=1}^3(n^2+2 n+1)\)[/tex]
- Sum: 29.
(e) [tex]\(\sum_{n=1}^{10} 5n\)[/tex]
- Sum: 275.
(f) [tex]\(\sum_{n=5}^{10} n^2\)[/tex]
- Sum: 355.
(g) [tex]\(\sum_{n=3}^8(n^2-2)\)[/tex]
- Sum: 187.
(h) [tex]\(\sum_{n=1}^5 \frac{2n+1}{n}\)[/tex]
- Sum: 12.283333333333333.
(i) [tex]\(\sum_{n=0}^4 \frac{n}{n+1}\)[/tex]
- Sum: 2.716666666666667.
### 4. Write the given series using symbol [tex]\(\sum\)[/tex]:
(a) [tex]\(5 + 7 + 9 + 11 + \ldots + 21\)[/tex]
- Series: [tex]\(\sum_{n=0}^{8}(5 + 2n)\)[/tex].
(b) [tex]\(2 + 4 + 6 + 8 + 10 + 12\)[/tex]
- Series: [tex]\(\sum_{n=1}^{6}(2n)\)[/tex].
(c) [tex]\(30 + 25 + 20 + \ldots + 5\)[/tex]
- Series: [tex]\(\sum_{n=0}^{5}(30 - 5n)\)[/tex].
(d) [tex]\(1 + 5 + 9 + 13 + 17 + 21\)[/tex]
- Series: [tex]\(\sum_{n=0}^{5}(1 + 4n)\)[/tex].
(e) [tex]\(1 + 4 + 9 + 16\)[/tex]
- Series: [tex]\(\sum_{n=1}^{4}(n^2)\)[/tex].
(f) [tex]\(a + ab^1 + ab^2 + ab^3 + ab^4\)[/tex]
- Series: [tex]\(\sum_{n=0}^{4}(ab^n)\)[/tex].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
