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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].
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