Question

Evaluate S5 for 300 + 150 + 75 + … and select the correct answer below. 18.75 93.75 581.25 145.3125

Answer 1

we have that

`300 + 150 + 75 +...`

Let

`a1=300\\ a2=150\\ a3=75`

we know that

`(a2)/(a1) =(150)/(300) \\\\ (a2)/(a1)=0.5 \\ \\ a2=a1*0.50`

`(a3)/(a2) =(75)/(150) \\\\ (a3)/(a2)=0.5 \\ \\ a3=a2*0.50`

so

`a(n+1)=an*0.50`

Is a geometric sequence

Find the value of
`a4`

`a(4)=a3*0.50`

`a(4)=75*0.50`

`a(4)=37.5`

Find the value of
`a5`

`a(5)=a4*0.50`

`a(5)=37.5*0.50`

`a(5)=18.75`

Find
`S5`

`S5=a1+a2+a3+a4+a5\\ S5=300+150+75+37.5+18.75\\ S5=581.25`

therefore

the answer is

`581.25`

Alternative Method

Applying the formula

`S_n=(a_1 (1-r^n))/(1-r) \\\\a_1=300 \\ r=(1)/(2)\\\\ S_5=(300(1-((1)/(2))^5))/(1-(1)/(2))\\\\=(300(1-(1)/(32)))/((1)/(2))\\\\=(300 * (31)/(32))/((1)/(2))\\\\=(75 * (31)/(8))/((1)/(2))\\\\=((2325)/(8))/((1)/(2))\\\\=(2325)/(8) * 2\\\\=(2325)/(4)\\\\=581 (1)/(4)\\\\=581.25`

therefore

the answer is

`581.25`

Answer 2

It's a geometric sequence.


`300, 150, 75,... \\ \\ a_1=300 \\ r=(a_2)/(a_1)=(150)/(300)=(1)/(2) \\ \\ S_n=(a_1 (1-r^n))/(1-r) \\ \Downarrow \\ S_5=(300(1-((1)/(2))^5))/(1-(1)/(2))=(300(1-(1)/(32)))/((1)/(2))=(300 * (31)/(32))/((1)/(2))=(75 * (31)/(8))/((1)/(2))=((2325)/(8))/((1)/(2))=(2325)/(8) * 2= \\ =(2325)/(4)=581 (1)/(4)=581.25`

The answer is 581.25.