Resources for AP and GP Problems
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Questions
15
With Worked SolutionClick Here -
Video Tutorials
4
Click Here -
HSC Questions
4
With Worked SolutionClick Here
AP and GP Problems Theory
![In solving AP and GP problems it is necessary to know all the formulae:\\ $\begin{aligned}[t] \text { A.P. } \quad T_n & =a+(n-1) d \\ S_n & =\frac{n}{2}(2 a+(n-1) d) \\ S_n & =\frac{n}{2}(a+l) \end{aligned}$\\ \hfill \linebreak $\begin{aligned}[t] G.P. \quad T_n&=a r^{n-1} \\ S_n&=\frac{a\left(r^n-1\right)}{r-1} \quad r>1 \\ S_n&=\frac{a\left(1-r^n\right)}{1-r} \quad r<1 \\ S_{\infty}&=\frac{a}{1-r} \quad-1<r<1 \end{aligned}$\\ \begin{multicols}{2} \textbf{Example 1}\\ %25848 If \(4,x,y\) form a G.P. and \(x,y,12\) form an A.P. then the values of \(x\) and \(y\) are?\\ \textbf{Example 1 solution}\\ \(\begin{aligned} r=\frac{y}{x}, r=\frac{x}{4} \rightarrow \frac{y}{x}=\frac{x}{4} \rightarrow x^{2}=4 y\ldots (1)\qquad\qquad & \end{aligned}\)\\ \(\begin{aligned} d=12-y, d=y-x \\ \rightarrow 12-y=y-x \rightarrow 2 y-x&=12\ldots (2) \end{aligned}\)\\ \(\begin{aligned} \text { In (2) } 4 y=2 x+24 \\ \rightarrow \text { sub into (1) } & x^{2}=2 x+24\\ &(x-6)(x+4)=0 \\ &\therefore x=6 \,\text { or }\, x=-4 \end{aligned}\)\\ \(\begin{aligned} \text{In (1) }\quad 4 y=36 \quad 4 y&=16\\ y=9\qquad y&=4 \end{aligned}\)\\ \(\begin{aligned} \therefore x=6, y=9 \quad \text { OR }\quad x=-4, y&=4 \end{aligned}\)\\ \columnbreak \textbf{Example 2}\\ %24662 A plant is 100 cm high when first measured. In the first month of observation it grows 20 cm and in each succeeding month the growth is 50\% of the previous week's growth. If this pattern continues, what will be the plant's ultimate height?\\ \textbf{Example 2 solution}\\ \(\begin{aligned} 20, & 10,5, \ldots \\ a=& 20, r=\frac{1}{2} \end{aligned}\)\\ \(\begin{aligned} S_{\infty} &=\frac{a}{1-r} \\ &=\frac{20}{1-\frac{1}{2}} \\ &=40 \end{aligned}\)\\ \(\begin{aligned} \therefore h &=140 \mathrm{~cm} \end{aligned}\)\\ \end{multicols}](/media/wwnb01rb/4764.png)