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Certainly! Let's solve the given question step-by-step.
### Part A: List the next four terms in the sequence.
The given sequence is [tex]\( -\frac{1}{5}, \frac{2}{6}, -\frac{3}{7}, \frac{4}{8}, \ldots \)[/tex]
By examining the pattern:
- Numerator: The numerals alternate in sign and increase by 1 each step.
- Denominator: The denominators form an increasing arithmetic sequence increasing by 1 each step and starting from 5.
Following this pattern, the next four terms can be determined as:
1. The 5th term: Numerator is [tex]\(-5\)[/tex] (since numerators alternate in sign). Denominator is [tex]\(5 + 4 = 9\)[/tex].
Therefore, the 5th term is [tex]\(-\frac{5}{9}\)[/tex].
2. The 6th term: Numerator is [tex]\(6\)[/tex]. Denominator is [tex]\(6 + 4 = 10\)[/tex].
Therefore, the 6th term is [tex]\(\frac{6}{10}\)[/tex].
3. The 7th term: Numerator is [tex]\(-7\)[/tex]. Denominator is [tex]\(7 + 4 = 11\)[/tex].
Therefore, the 7th term is [tex]\(-\frac{7}{11}\)[/tex].
4. The 8th term: Numerator is [tex]\(8\)[/tex]. Denominator is [tex]\(8 + 4 = 12\)[/tex].
Therefore, the 8th term is [tex]\(\frac{8}{12}\)[/tex].
Next four terms:
[tex]\[ -\frac{5}{9}, \frac{6}{10}, -\frac{7}{11}, \frac{8}{12} \][/tex]
### Part B: Write the explicit equation for [tex]\( f(n) \)[/tex] to represent the sequence.
Let’s determine the explicit equation [tex]\( f(n) \)[/tex]:
For the numerators:
- When [tex]\( n \)[/tex] is odd, the numerators are negative: [tex]\(-n\)[/tex]
- When [tex]\( n \)[/tex] is even, the numerators are positive: [tex]\(n\)[/tex]
Using the properties of powers of [tex]\(-1\)[/tex]:
[tex]\[ (-1)^{n+1} \][/tex]
This expression will be [tex]\( -1 \)[/tex] if [tex]\( n \)[/tex] is odd (since the power [tex]\( n+1 \)[/tex] will be even) and [tex]\( 1 \)[/tex] if [tex]\( n \)[/tex] is even.
For the denominators:
- Each denominator is [tex]\( n + 4 \)[/tex]
Putting it all together, the explicit formula for [tex]\( f(n) \)[/tex] is:
[tex]\[ f(n) = \frac{(-1)^{n+1} \cdot n}{n + 4} \][/tex]
### Part C: Is the sign of [tex]\( f(56) \)[/tex] positive or negative?
We need to determine the sign of [tex]\( f(56) \)[/tex] without calculating the value of [tex]\( f(56) \)[/tex].
Recall:
[tex]\[ f(n) = \frac{(-1)^{n+1} \cdot n}{n + 4} \][/tex]
Examine the expression [tex]\( (-1)^{n+1} \)[/tex]:
- For [tex]\( n = 56 \)[/tex]:
[tex]\[ (-1)^{56 + 1} = (-1)^{57} \][/tex]
Since 57 is an odd number, [tex]\( (-1)^{57} \)[/tex] is [tex]\(-1\)[/tex].
So, the sign of [tex]\( f(56) \)[/tex] is determined by the factor [tex]\( (-1)^{57} \)[/tex], which is [tex]\(-1\)[/tex]. Hence, the sign of [tex]\( f(56) \)[/tex] is negative.
Thus, [tex]\( f(56) \)[/tex] is negative.
Therefore:
[tex]\[ \boxed{-0.5555555555555556, 0.6, -0.6363636363636364, 0.6666666666666666, \text{The sign of } f(56) \text{ is negative}} \][/tex]
### Part A: List the next four terms in the sequence.
The given sequence is [tex]\( -\frac{1}{5}, \frac{2}{6}, -\frac{3}{7}, \frac{4}{8}, \ldots \)[/tex]
By examining the pattern:
- Numerator: The numerals alternate in sign and increase by 1 each step.
- Denominator: The denominators form an increasing arithmetic sequence increasing by 1 each step and starting from 5.
Following this pattern, the next four terms can be determined as:
1. The 5th term: Numerator is [tex]\(-5\)[/tex] (since numerators alternate in sign). Denominator is [tex]\(5 + 4 = 9\)[/tex].
Therefore, the 5th term is [tex]\(-\frac{5}{9}\)[/tex].
2. The 6th term: Numerator is [tex]\(6\)[/tex]. Denominator is [tex]\(6 + 4 = 10\)[/tex].
Therefore, the 6th term is [tex]\(\frac{6}{10}\)[/tex].
3. The 7th term: Numerator is [tex]\(-7\)[/tex]. Denominator is [tex]\(7 + 4 = 11\)[/tex].
Therefore, the 7th term is [tex]\(-\frac{7}{11}\)[/tex].
4. The 8th term: Numerator is [tex]\(8\)[/tex]. Denominator is [tex]\(8 + 4 = 12\)[/tex].
Therefore, the 8th term is [tex]\(\frac{8}{12}\)[/tex].
Next four terms:
[tex]\[ -\frac{5}{9}, \frac{6}{10}, -\frac{7}{11}, \frac{8}{12} \][/tex]
### Part B: Write the explicit equation for [tex]\( f(n) \)[/tex] to represent the sequence.
Let’s determine the explicit equation [tex]\( f(n) \)[/tex]:
For the numerators:
- When [tex]\( n \)[/tex] is odd, the numerators are negative: [tex]\(-n\)[/tex]
- When [tex]\( n \)[/tex] is even, the numerators are positive: [tex]\(n\)[/tex]
Using the properties of powers of [tex]\(-1\)[/tex]:
[tex]\[ (-1)^{n+1} \][/tex]
This expression will be [tex]\( -1 \)[/tex] if [tex]\( n \)[/tex] is odd (since the power [tex]\( n+1 \)[/tex] will be even) and [tex]\( 1 \)[/tex] if [tex]\( n \)[/tex] is even.
For the denominators:
- Each denominator is [tex]\( n + 4 \)[/tex]
Putting it all together, the explicit formula for [tex]\( f(n) \)[/tex] is:
[tex]\[ f(n) = \frac{(-1)^{n+1} \cdot n}{n + 4} \][/tex]
### Part C: Is the sign of [tex]\( f(56) \)[/tex] positive or negative?
We need to determine the sign of [tex]\( f(56) \)[/tex] without calculating the value of [tex]\( f(56) \)[/tex].
Recall:
[tex]\[ f(n) = \frac{(-1)^{n+1} \cdot n}{n + 4} \][/tex]
Examine the expression [tex]\( (-1)^{n+1} \)[/tex]:
- For [tex]\( n = 56 \)[/tex]:
[tex]\[ (-1)^{56 + 1} = (-1)^{57} \][/tex]
Since 57 is an odd number, [tex]\( (-1)^{57} \)[/tex] is [tex]\(-1\)[/tex].
So, the sign of [tex]\( f(56) \)[/tex] is determined by the factor [tex]\( (-1)^{57} \)[/tex], which is [tex]\(-1\)[/tex]. Hence, the sign of [tex]\( f(56) \)[/tex] is negative.
Thus, [tex]\( f(56) \)[/tex] is negative.
Therefore:
[tex]\[ \boxed{-0.5555555555555556, 0.6, -0.6363636363636364, 0.6666666666666666, \text{The sign of } f(56) \text{ is negative}} \][/tex]
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