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

Respuesta :

naǫ
It's a geometric sequence.

[tex]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[/tex]

The answer is 581.25.
calculista

we have that

[tex] 300 + 150 + 75 +... [/tex]

Let

[tex] a1=300\\ a2=150\\ a3=75 [/tex]

we know that

[tex] \frac{a2}{a1} =\frac{150}{300} \\\\ \frac{a2}{a1}=0.5 \\ \\ a2=a1*0.50 [/tex]

[tex] \frac{a3}{a2} =\frac{75}{150} \\\\ \frac{a3}{a2}=0.5 \\ \\ a3=a2*0.50 [/tex]

so

[tex] a(n+1)=an*0.50 [/tex]

Is a geometric sequence

Find the value of [tex] a4 [/tex]

[tex] a(4)=a3*0.50 [/tex]

[tex] a(4)=75*0.50 [/tex]

[tex] a(4)=37.5 [/tex]

Find the value of [tex] a5 [/tex]

[tex] a(5)=a4*0.50 [/tex]

[tex] a(5)=37.5*0.50 [/tex]

[tex] a(5)=18.75 [/tex]

Find [tex] S5 [/tex]

[tex] S5=a1+a2+a3+a4+a5\\ S5=300+150+75+37.5+18.75\\ S5=581.25 [/tex]

therefore

the answer is

[tex] 581.25 [/tex]

Alternative Method

Applying the formula

[tex] 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 [/tex]

therefore

the answer is

[tex] 581.25 [/tex]

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