To find the sum of the first five terms of a geometric sequence with the given initial term [tex]\( a_1 = 5 \)[/tex] and the common ratio [tex]\( r = \frac{1}{5} \)[/tex], we will use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric sequence:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Here, [tex]\( n = 5 \)[/tex], [tex]\( a_1 = 5 \)[/tex], and [tex]\( r = \frac{1}{5} \)[/tex].
Let's plug the values into the formula step by step.
1. Calculate [tex]\( r^5 \)[/tex]:
[tex]\[ r = \frac{1}{5} \][/tex]
[tex]\[ r^5 = \left(\frac{1}{5}\right)^5 = \frac{1}{5^5} = \frac{1}{3125} \][/tex]
2. Calculate [tex]\( 1 - r^5 \)[/tex]:
[tex]\[ 1 - r^5 = 1 - \frac{1}{3125} = \frac{3125}{3125} - \frac{1}{3125} = \frac{3124}{3125} \][/tex]
3. Calculate [tex]\( 1 - r \)[/tex]:
[tex]\[ 1 - r = 1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5} \][/tex]
4. Substitute these values back into the sum formula:
[tex]\[ S_5 = 5 \cdot \frac{\frac{3124}{3125}}{\frac{4}{5}} \][/tex]
5. Simplify the fraction:
[tex]\[ S_5 = 5 \cdot \frac{3124}{3125} \cdot \frac{5}{4} \][/tex]
[tex]\[ S_5 = 5 \cdot \frac{3124 \cdot 5}{3125 \cdot 4} \][/tex]
[tex]\[ S_5 = 5 \cdot \frac{15620}{12500} \][/tex]
[tex]\[ S_5 = 5 \cdot 1.2496 \][/tex]
[tex]\[ S_5 = 6.248 \][/tex]
Therefore, the sum of the first five terms of the geometric sequence is [tex]\( 6.248 \)[/tex].
So, the sum of the first five terms of the geometric sequence with [tex]\( a_1 = 5 \)[/tex] and [tex]\( r = \frac{1}{5} \)[/tex] is [tex]\( 6.248 \)[/tex].
