Answer:
[tex]\bold{\dfrac{11}{2 }}[/tex]
Step-by-step explanation:
Given the geometric series:
[tex]\{\dfrac{1}2, -1, 2, -4, ..... \}[/tex]
To find:
Sum of series upto 5 terms using the geometric series formula = ?
Solution:
Formula for sum of a n terms of a geometric series is given as:
[tex]S_n=\dfrac{a(1-r^n)}{1-r} \ \{r<1 \}[/tex]
[tex]a[/tex] is the first term of the geometric series
[tex]r[/tex] is the common ratio between each term (2nd term divided by 1st term or 3rd term divided by 2nd term ..... ).
Here:
[tex]a = \dfrac{1}{2}[/tex]
[tex]r = \dfrac{-1}{\dfrac{1}{2}} = -2[/tex]
[tex]n=5[/tex]
So, applying the formula for given values:
[tex]S_5=\dfrac{\dfrac{1}2(1-(-2)^5)}{1-(-2)} \\\Rightarrow S_5=\dfrac{1-(-32)}{2 \times 3} \\\Rightarrow S_5=\dfrac{1+32}{6} \\\Rightarrow S_5=\dfrac{33}{6} \\\Rightarrow \bold{S_5=\dfrac{11}{2}}[/tex]
So, the answer is
[tex]\bold{\dfrac{11}{2 }}[/tex]
