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Sagot :
Let's analyze the given series and the multiple-choice options to identify which option corresponds with the series provided.
Given series:
[tex]\[ 2.5 + 2.5(1.2) + 2.5(1.2)^2 + \cdots + 2.5(1.2)^{87} \][/tex]
This series can be rewritten by factoring out the 2.5:
[tex]\[ 2.5 \left( 1 + (1.2) + (1.2)^2 + \cdots + (1.2)^{87} \right) \][/tex]
We recognize that this is a geometric series with the first term [tex]\(a = 1\)[/tex] and the common ratio [tex]\(r = 1.2\)[/tex]. The number of terms in the series is 88, starting from [tex]\((1.2)^0\)[/tex] to [tex]\((1.2)^{87}\)[/tex].
Now, let's match this with the given multiple-choice options.
Option A:
[tex]\[ \sum_{k=1}^{87} 2.5(1.2)^k \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=87\)[/tex], corresponding to [tex]\(2.5(1.2)^1\)[/tex] to [tex]\(2.5(1.2)^{87}\)[/tex]. Our given series includes the term [tex]\(2.5\)[/tex], i.e., [tex]\(2.5(1.2)^0\)[/tex], thus option A misses this initial term.
Option B:
[tex]\[ \sum_{k=1}^{87} 2.5(1.2)^{k-1} \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=87\)[/tex], translating to [tex]\(2.5(1.2)^0\)[/tex] to [tex]\(2.5(1.2)^{86}\)[/tex]. This range starts in the right place but does not reach [tex]\(2.5(1.2)^{87}\)[/tex]; it stops one term short of our series.
Option C:
[tex]\[ \sum_{k=1}^{88} 2.5(1.2)^k \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=88\)[/tex], translating to [tex]\(2.5(1.2)^1\)[/tex] to [tex]\(2.5(1.2)^{88}\)[/tex]. This range starts one step further than our given series and includes an extra term [tex]\(2.5(1.2)^{88}\)[/tex].
Option D:
[tex]\[ \sum_{k=1}^{88} 2.5(1.2)^{k-1} \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=88\)[/tex], translating to [tex]\(2.5(1.2)^0\)[/tex] to [tex]\(2.5(1.2)^{87}\)[/tex]. This perfectly matches our series, starting from [tex]\(2.5(1.2)^0\)[/tex] and ending at [tex]\(2.5(1.2)^{87}\)[/tex].
Based on the step-by-step analysis, the correct option that represents the given series is:
[tex]\[ \boxed{\sum_{k=1}^{88} 2.5(1.2)^{k-1}} \][/tex]
This corresponds to option D. Therefore, the correct answer is:
D. [tex]\(\sum_{k=1}^{88} 2.5(1.2)^{k-1}\)[/tex]
Given series:
[tex]\[ 2.5 + 2.5(1.2) + 2.5(1.2)^2 + \cdots + 2.5(1.2)^{87} \][/tex]
This series can be rewritten by factoring out the 2.5:
[tex]\[ 2.5 \left( 1 + (1.2) + (1.2)^2 + \cdots + (1.2)^{87} \right) \][/tex]
We recognize that this is a geometric series with the first term [tex]\(a = 1\)[/tex] and the common ratio [tex]\(r = 1.2\)[/tex]. The number of terms in the series is 88, starting from [tex]\((1.2)^0\)[/tex] to [tex]\((1.2)^{87}\)[/tex].
Now, let's match this with the given multiple-choice options.
Option A:
[tex]\[ \sum_{k=1}^{87} 2.5(1.2)^k \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=87\)[/tex], corresponding to [tex]\(2.5(1.2)^1\)[/tex] to [tex]\(2.5(1.2)^{87}\)[/tex]. Our given series includes the term [tex]\(2.5\)[/tex], i.e., [tex]\(2.5(1.2)^0\)[/tex], thus option A misses this initial term.
Option B:
[tex]\[ \sum_{k=1}^{87} 2.5(1.2)^{k-1} \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=87\)[/tex], translating to [tex]\(2.5(1.2)^0\)[/tex] to [tex]\(2.5(1.2)^{86}\)[/tex]. This range starts in the right place but does not reach [tex]\(2.5(1.2)^{87}\)[/tex]; it stops one term short of our series.
Option C:
[tex]\[ \sum_{k=1}^{88} 2.5(1.2)^k \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=88\)[/tex], translating to [tex]\(2.5(1.2)^1\)[/tex] to [tex]\(2.5(1.2)^{88}\)[/tex]. This range starts one step further than our given series and includes an extra term [tex]\(2.5(1.2)^{88}\)[/tex].
Option D:
[tex]\[ \sum_{k=1}^{88} 2.5(1.2)^{k-1} \][/tex]
This option sums from [tex]\(k=1\)[/tex] to [tex]\(k=88\)[/tex], translating to [tex]\(2.5(1.2)^0\)[/tex] to [tex]\(2.5(1.2)^{87}\)[/tex]. This perfectly matches our series, starting from [tex]\(2.5(1.2)^0\)[/tex] and ending at [tex]\(2.5(1.2)^{87}\)[/tex].
Based on the step-by-step analysis, the correct option that represents the given series is:
[tex]\[ \boxed{\sum_{k=1}^{88} 2.5(1.2)^{k-1}} \][/tex]
This corresponds to option D. Therefore, the correct answer is:
D. [tex]\(\sum_{k=1}^{88} 2.5(1.2)^{k-1}\)[/tex]
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