Sum of GP members

Determine the sum of the GP 30, 6, 1.2, to 5 terms. What is the sum of all terms (to infinity)?

Correct answer:

s₅ =  37.488
s =  37.5

Step-by-step explanation:

$$\begin{gathered} a_1=30 \\ {a}_2=6 \\ {a}_3=1.2 \\ r=a_2\mathrm{/}{a}_1=6\mathrm{/}30={\displaystyle \frac{1}{5}}=0.2 \\ {r}_2=a_3\mathrm{/}{a}_2=1.2\mathrm{/}6={\displaystyle \frac{1}{5}}=0.2 \\ r=r_2 \\ {a}_4=a_3\cdot r=1.2\cdot 0.2={\displaystyle \frac{6}{25}}=0.24 \\ {a}_5=a_4\cdot r=0.24\cdot 0.2={\displaystyle \frac{6}{125}}=0.048 \\ {s}_5=a_1+a_2+a_3+a_4+a_5=30+6+1.2+0.24+0.048={\displaystyle \frac{4686}{125}}=37\frac{61}{125}=37.488 \end{gathered}$$

$s=a_1\cdot {\displaystyle \frac{1}{1-r}}=30\cdot {\displaystyle \frac{1}{1-0.2}}={\displaystyle \frac{75}{2}}=37\frac{1}{2}=37.5$

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