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NSW Y12 Maths Standard 1 NSW Y12 Maths Standard 2 NSW Y12 Maths Advanced NSW Y12 Maths Extension 1 NSW Y12 Maths Extension 2

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NSW Y12 Maths - Advanced Series and Finance Applications of APs and GPs

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Applications of APs and GPs Theory

For the applications the following formulas are needed:\\ $\begin{aligned} \text { A.P. } \quad & T_n=a+(n-1) d \\ S_n & =\frac{n}{2}(a+l) \\ S_n & =\frac{n}{2}(2 a+(n-1) d) \\ T_n & =a r^{n-1} \\ S_n & =\frac{a\left(n^n-1\right)}{r-1}, n>1 \\ \text { G.P } & =a \frac{\left(1-r^n\right)}{1-r}, v<1 \\ S_n & =\frac{a}{1-r}-1<n<1 \end{aligned}$ \begin{multicols}{2} \textbf{Example 1}\\ How many times does a clock chime in 12 hours if it strikes once for each half hour, and once at 1 o'clock, twice at 1 o'clock etc.?\\ \textbf{Example 1 solution}\\ The clock will strike 12 times for each half hour. On the hour \(1+2+3+\cdots+12\) $\begin{aligned} \therefore \quad c_{12} & =\frac{12}{2}(1+12) \\ & =6 \times 13 \\ & =78 \\ \therefore \text { Total } & =78+12 \\ & =90 \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ An object falls \(0.5 \mathrm{~m}\) in the first second. then, each second after, it falls \(\dfrac{5}{6}\) of its previous fall. Find how far it will fall.\\ \textbf{Example 2 solution}\\ $\begin{aligned} &\text { This is a G.P. } \\ & 0.5+0.5 \times \frac{5}{6}+0.5 \times \frac{5}{6} \times \frac{5}{6} \times \cdots \\ &a=0.5 \quad r=\frac{5}{6} \\ S_{\infty} & =\frac{a}{1-r} \\ &= \frac{1}{2} \div\left(1-\frac{5}{6}\right) \\ & =3 \mathrm{~m} \end{aligned}$\\ \end{multicols}

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