Find the sum of this infinite geometric series where a1=0.3 and r=0.55

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Question:

Find the sum of this infinite geometric series where a1=0.3 and r=0.55

Answers

Hello, Marymack!We know that in a geometric sequence defined by [tex]\mathsf{a_1}[/tex] and [tex]\mathsf{r}[/tex] , the sum can be calculated the following way:[tex]\mathsf{S_n = \dfrac{a_1}{1-r}}\\ \\ \\ \\ \mathsf{S_n = \dfrac{0.3}{1-0.55}}\\ \\ \\ \mathsf{S_n = \dfrac{0.3}{0.45}}\\ \\ \\ \boxed{\mathsf{S_n = \dfrac{2}{3}=0.666...}}[/tex]

Answer:The sum of the infinite geometric sequence is given by:[tex]S_{\infty} = \frac{a_1}{1-r}[/tex]          ....[1]where,[tex]a_1[/tex] is the first termr is the common ratio.As per the statement:Given that [tex]a_1 = 0.3[/tex] and r = 0.55To find the sum of this infinite geometric series.Substitute the given values in [1] we have;[tex]S_{\infty} = \frac{0.3}{1-0.55}[/tex]⇒[tex]S_{\infty} = \frac{0.3}{0.45}[/tex]Simplify:[tex]S_{\infty} \approx 0.67[/tex]Therefore,  the sum of this infinite geometric series approximate is, 0.67

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