Question

Find the sum of the first five terms using the geometric series formula for the sequence left curly bracket 1 half comma space minus 1 comma 2 comma negative 4 comma... right curly bracket

Answer 1

Answer:

$\bold{\dfrac{11}{2 }}$

Step-by-step explanation:

Given the geometric series:

$\{\dfrac{1}2, -1, 2, -4, ..... \}$

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:

$S_n=\dfrac{a(1-r^n)}{1-r} \ \{r$

$a$ is the first term of the geometric series

$r$ is the common ratio between each term (2nd term divided by 1st term or 3rd term divided by 2nd term ..... ).

Here:

$a = \dfrac{1}{2}$

$r = \dfrac{-1}{\dfrac{1}{2}} = -2$

$n=5$

So, applying the formula for given values:

$S_5=\dfrac{\dfrac{1}2(1-(-2)^5)}{1-(-2)} \\\Rightarrow S_5=\dfrac{1-(-32)}{2 \times 3} \\\Rightarrow S_5=\dfrac{1+32}{6} \\\Rightarrow S_5=\dfrac{33}{6} \\\Rightarrow \bold{S_5=\dfrac{11}{2}}$

So, the answer is

$\bold{\dfrac{11}{2 }}$