Answer :
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]
[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]
