$$ \large \displaystyle S_{\infty} = \frac{\text{first term}}{1-\text{common ratio}} $$
Question 1
Find the first three terms of the geometric series for which the common ratio \( r=0.6 \) and \( S_{\infty} = 25 \).
\( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ \frac{a}{1-0.6} &= 25 \\ \frac{a}{0.4} &= 25 \\ a &= 25 \times 0.4 \\ a &= 10 \\ T_1 &= 10 \\ T_2 &= 10 \times 0.6 \\ T_3 &= 10 \times 0.6^2 \\ \therefore 10, 6 &\text{ and } 3.6 \end{align} \)
Question 2
Find the first three terms of the geometric series for which the first term \( a=10 \) and \( S_{\infty} = 120 \).
\( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ \frac{48}{1-r} &= 120 \\ \frac{1-r}{48} &= \frac{1}{120} \\ 1-r &= \frac{48}{120} \\ -r &= 0.4-1 \\ -r &= -0.6 \\ r &= 0.6 \\ T_1 &=48 \\ T_2 &= 48 \times 0.6 \\ T_3 &= 48 \times 0.6^2 \\ \therefore 48, 28.8 &\text{ and } 17.28 \end{align} \)
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