Question

Need help with geometric series'!
Please include a step by step explanation.

Answer 1

`- 13107`

Explanation:

We would like to evaluate the following Geometric Series ,

`\displaystyle\longrightarrow \sum _(n=1)^8 (-4)^(n-1 )`

In a geometric series, a common number is multiplied to the previous term in order to find the next term . And that common number is called common ratio (r) .

As we know that ,

`\displaystyle\longrightarrow \sum_(x = 1)^n f(x) = f(1) + f(2) + \dots + f(n)`

So , we can write the series as ,

`\displaystyle\longrightarrow (-4)^(1-1)+(-4)^(2-1)+\dots +(-4)^(8-1)`

Simplify,

`\displaystyle\longrightarrow (-4)^0 + (-4)^1+(-4)^2+\dots +(-4)^7`

We can find the common ratio by dividing and successive term by its preceding term , as ;

`\displaystyle\longrightarrow r =(-4)/(1)=-4`

Again , here ;

• First term = a =
  `(-4)^0=1`
• Common ratio =
  `-4`
• number of terms (n) = 8

And we can find the sum of geometric series using the formula , ( this is used when the value of r is less than 1 , here it is -4 ) .

`\displaystyle\longrightarrow \Bigg[ Sum = (a(1-r^n))/(1-r)\Bigg]`

• where the symbols have their usual meaning.

On substituting the respective values, we have;

`\displaystyle \longrightarrow\rm{ Sum }=\frac{ 1\{1-(-4)^8\}}{1-(-4)}\\`

Simplify ,

`\displaystyle\longrightarrow \rm{Sum} = (1-65536)/(1+4)\\`

`\displaystyle\longrightarrow \rm{Sum} = (-65535)/(5)`

Simplify by dividing ,

`\displaystyle\longrightarrow \underline{\underline{ \rm{Sum} = -13107}}`

And we are done !