Question

Please help meeeeeeeeeee

Answer 1

`(468)/(25)`

Explanation:

We are asked to find

`\displaystyle \sum _(n=0)^33\left((1)/(5)\right)^(n-1)`

We are asked to find s₄ which is the sum of the first 4 terms of the series. Since n starts at 0, this would be
a₀ + a₁ + a₂ + a₃

We can compute the individual terms and add them up

`n = 0 \\\rightarrow \displaystyle a_0 = 3\left((1)/(5)\right)^(0-1)\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(-1)\\\\\rightarrow \displaystyle 3\left(5}\right)^(1)\\ \rightarrow \displaystyle 15`

This would be the first term of the geometric sequence

n = 1

`\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(1-1)\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(0)\\\\\rightarrow \displaystyle 3\cdot 1}\\ \rightarrow \displaystyle a_1 = 3`

n = 2

`\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(2-1)\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(1)\\\\\rightarrow \displaystyle a_2 = (3)/(5)`

n = 3

`\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(3-1)\\\rightarrow \displaystyle 3\left((1)/(5)\right)^(2)\\\\\rightarrow \displaystyle a_3 = (3)/(25)`

Adding these we get

15 + 3 + 3/5 + 3/25 =
`(468)/(25)`

Alternate route
In a geometric sequence the sum of the formula for the first n terms will be

`a_0 + a_1(1-r^(n))/(1-r)`

We computed a₀ as 15 and a₁ as 3

So sum

`= 15 + 3\left((1-\left((1)/(5)\right)^3)/(1-(1)/(5))\right)\\`

If you evaluate this, it should work out to the same fraction:

`(468)/(25)`