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
\(581.25\)
