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1. Use the recursive formula to find the first five terms in the arithmetic sequence.

Given:
[tex]\[ f(1) = \frac{1}{5} \][/tex]
[tex]\[ f(n) = f(n-1) + \frac{1}{5} \][/tex]

A. [tex]\(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1\)[/tex]
B. [tex]\(\frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1, 1\frac{1}{5}\)[/tex]
C. [tex]\(\frac{1}{5}, 0, -\frac{1}{5}, -\frac{2}{5}, -\frac{3}{5}\)[/tex]
D. [tex]\(0, -\frac{1}{5}, -\frac{2}{5}, -\frac{3}{5}, -\frac{4}{5}\)[/tex]

Sagot :

GinnyAnswer
Sure, let's find the first five terms in the arithmetic sequence using the given recursive formula step-by-step. The recursive formula is defined as follows:

[tex]\[ \begin{aligned} f(1) & = \frac{1}{5}, \\ f(n) & = f(n-1) + \frac{1}{5}. \end{aligned} \][/tex]

Here are the steps to find the first five terms:

1. First term: [tex]\( f(1) = \frac{1}{5} \)[/tex].
2. Second term: We use the formula [tex]\( f(n) = f(n-1) + \frac{1}{5} \)[/tex]. Substituting [tex]\( n = 2 \)[/tex]:
[tex]\[ f(2) = f(1) + \frac{1}{5} = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}. \][/tex]
3. Third term: Substituting [tex]\( n = 3 \)[/tex]:
[tex]\[ f(3) = f(2) + \frac{1}{5} = \frac{2}{5} + \frac{1}{5} = \frac{3}{5}. \][/tex]
4. Fourth term: Substituting [tex]\( n = 4 \)[/tex]:
[tex]\[ f(4) = f(3) + \frac{1}{5} = \frac{3}{5} + \frac{1}{5} = \frac{4}{5}. \][/tex]
5. Fifth term: Substituting [tex]\( n = 5 \)[/tex]:
[tex]\[ f(5) = f(4) + \frac{1}{5} = \frac{4}{5} + \frac{1}{5} = 1. \][/tex]

Thus, the first five terms of the sequence are:
[tex]\[ \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1. \][/tex]
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