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

`\bold{(11)/(2 )}`

Explanation:

Given the geometric series:

`\{\frac{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=(a(1-r^n))/(1-r) \ \{r<1 \}`

`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 = (1)/(2)`

`r = (-1)/((1)/(2)) = -2`

`n=5`

So, applying the formula for given values:

`S_5=\frac{\frac{1}2(1-(-2)^5)}{1-(-2)} \\\Rightarrow S_5=(1-(-32))/(2 * 3) \\\Rightarrow S_5=(1+32)/(6) \\\Rightarrow S_5=(33)/(6) \\\Rightarrow \bold{S_5=(11)/(2)}`

So, the answer is

`\bold{(11)/(2 )}`