Question

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

Answer 1

It's a geometric sequence.

$300, 150, 75,... \\ \\ a_1=300 \\ r=\frac{a_2}{a_1}=\frac{150}{300}=\frac{1}{2} \\ \\ S_n=\frac{a_1 (1-r^n)}{1-r} \\ \Downarrow \\ S_5=\frac{300(1-(\frac{1}{2})^5)}{1-\frac{1}{2}}=\frac{300(1-\frac{1}{32})}{\frac{1}{2}}=\frac{300 \times \frac{31}{32}}{\frac{1}{2}}=\frac{75 \times \frac{31}{8}}{\frac{1}{2}}=\frac{\frac{2325}{8}}{\frac{1}{2}}=\frac{2325}{8} \times 2= \\ =\frac{2325}{4}=581 \frac{1}{4}=581.25$

The answer is 581.25.

Answer 2

we have that

$300 + 150 + 75 +...$

Let

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

we know that

$\frac{a2}{a1} =\frac{150}{300} \\\\ \frac{a2}{a1}=0.5 \\ \\ a2=a1*0.50$

$\frac{a3}{a2} =\frac{75}{150} \\\\ \frac{a3}{a2}=0.5 \\ \\ a3=a2*0.50$

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

$S_n=\frac{a_1 (1-r^n)}{1-r} \\\\a_1=300 \\ r=\frac{1}{2}\\\\ S_5=\frac{300(1-(\frac{1}{2})^5)}{1-\frac{1}{2}}\\\\=\frac{300(1-\frac{1}{32})}{\frac{1}{2}}\\\\=\frac{300 \times \frac{31}{32}}{\frac{1}{2}}\\\\=\frac{75 \times \frac{31}{8}}{\frac{1}{2}}\\\\=\frac{\frac{2325}{8}}{\frac{1}{2}}\\\\=\frac{2325}{8} \times 2\\\\=\frac{2325}{4}\\\\=581 \frac{1}{4}\\\\=581.25$

therefore

the answer is

$581.25$