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Let's analyze and solve each part of the question step-by-step.
### Step-by-Step Solution
#### Series Analysis
We have a series given as: [tex]\(27, +2, -9, +2, +3, +2, -1, \ldots\)[/tex].
We can notice that this series consists of two distinct parts that are interwoven:
1. A main series with some pattern: [tex]\(27, -9, 3, -1,\ldots\)[/tex]
2. A secondary series, interspersed with repetitive values of [tex]\(+2, +2, +2, \ldots\)[/tex]
#### 3.2.1 Determine the Value of Term 15
To find the 15th term in this series, let’s first identify the main series and the secondary series individually:
- The main series has elements: [tex]\(27, -9, 3, -1\)[/tex], which repeat periodically every four terms.
- The secondary series repeatedly adds [tex]\(+2\)[/tex] to every term.
To find the 15th term,
1. We determine the position in the main series: since these series repeat every 4 terms, the 15th position modulo 4 gives us the pattern position.
[tex]\[ (15 - 1) \mod 4 = 14 \mod 4 = 2 \][/tex]
So, the 15th term aligns with the 3rd term in the main series.
From the main series pattern: [tex]\(27, -9, 3, -1\)[/tex]
- The 3rd term is [tex]\(3\)[/tex].
To find the corresponding secondary series term for the 15th position,
- Since every position in the secondary series is [tex]\(+2\)[/tex],
[tex]\[ 2 \][/tex]
Thus, the value of term 15 combines these:
[tex]\[ \text{Term 15} = 3 + 2 = 5 \][/tex]
#### 3.2.2 Determine the Sum of the First 15 Terms
To find the sum of the first 15 terms, we need to sum up each term of the combined series step-by-step.
Since the main series pattern is [tex]\(27, -9, 3, -1\)[/tex] repeating every 4 terms, and the secondary term is always [tex]\(2\)[/tex], the pattern for the combined series over the first 15 terms can be broken down into sums of:
[tex]\[ \begin{aligned} & \text{1st term: } 27+2 = 29, \\ & \text{2nd term: } -9+2 = -7, \\ & \text{3rd term: } 3+2 = 5, \\ & \text{4th term: } -1+2 = 1. \\ \end{aligned} \][/tex]
This 4-term pattern repeats.
So, the combined series up to 15 terms follows this cycle and we sum these:
[tex]\[ \begin{aligned} & 4 \times (29 - 7 + 5 + 1) + \text{remaining 15%4 terms.} \end{aligned} \][/tex]
Calculating inside the brackets:
[tex]\[ 29 - 7 + 5 + 1 = 28 \][/tex]
Then, summing these results for cycles:
[tex]\[ 3 \times 28 + \text{sum of remaining terms: two cycles} = 84 + (29 + -7 +5 + 1 +29 - 7) \][/tex]
Calculating,
[tex]\[Sum of next 29 - 7 12+ 29 +(5-7 + ) \][/tex]
Adding calculated values
Therefore the sum of the first 15 terms is:
\( Sum of term 195
111
So Sollution above analysis and steps
Results are the term 15 th
}
### Step-by-Step Solution
#### Series Analysis
We have a series given as: [tex]\(27, +2, -9, +2, +3, +2, -1, \ldots\)[/tex].
We can notice that this series consists of two distinct parts that are interwoven:
1. A main series with some pattern: [tex]\(27, -9, 3, -1,\ldots\)[/tex]
2. A secondary series, interspersed with repetitive values of [tex]\(+2, +2, +2, \ldots\)[/tex]
#### 3.2.1 Determine the Value of Term 15
To find the 15th term in this series, let’s first identify the main series and the secondary series individually:
- The main series has elements: [tex]\(27, -9, 3, -1\)[/tex], which repeat periodically every four terms.
- The secondary series repeatedly adds [tex]\(+2\)[/tex] to every term.
To find the 15th term,
1. We determine the position in the main series: since these series repeat every 4 terms, the 15th position modulo 4 gives us the pattern position.
[tex]\[ (15 - 1) \mod 4 = 14 \mod 4 = 2 \][/tex]
So, the 15th term aligns with the 3rd term in the main series.
From the main series pattern: [tex]\(27, -9, 3, -1\)[/tex]
- The 3rd term is [tex]\(3\)[/tex].
To find the corresponding secondary series term for the 15th position,
- Since every position in the secondary series is [tex]\(+2\)[/tex],
[tex]\[ 2 \][/tex]
Thus, the value of term 15 combines these:
[tex]\[ \text{Term 15} = 3 + 2 = 5 \][/tex]
#### 3.2.2 Determine the Sum of the First 15 Terms
To find the sum of the first 15 terms, we need to sum up each term of the combined series step-by-step.
Since the main series pattern is [tex]\(27, -9, 3, -1\)[/tex] repeating every 4 terms, and the secondary term is always [tex]\(2\)[/tex], the pattern for the combined series over the first 15 terms can be broken down into sums of:
[tex]\[ \begin{aligned} & \text{1st term: } 27+2 = 29, \\ & \text{2nd term: } -9+2 = -7, \\ & \text{3rd term: } 3+2 = 5, \\ & \text{4th term: } -1+2 = 1. \\ \end{aligned} \][/tex]
This 4-term pattern repeats.
So, the combined series up to 15 terms follows this cycle and we sum these:
[tex]\[ \begin{aligned} & 4 \times (29 - 7 + 5 + 1) + \text{remaining 15%4 terms.} \end{aligned} \][/tex]
Calculating inside the brackets:
[tex]\[ 29 - 7 + 5 + 1 = 28 \][/tex]
Then, summing these results for cycles:
[tex]\[ 3 \times 28 + \text{sum of remaining terms: two cycles} = 84 + (29 + -7 +5 + 1 +29 - 7) \][/tex]
Calculating,
[tex]\[Sum of next 29 - 7 12+ 29 +(5-7 + ) \][/tex]
Adding calculated values
Therefore the sum of the first 15 terms is:
\( Sum of term 195
111
So Sollution above analysis and steps
Results are the term 15 th
}
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