Question

in a garden there are 162 mango trees which are arranged as 30 mango trees in the first row, 27 in the second, 24 in the thirdb and so on How many trees are there in the last row. How many rows are there of mango trees.

Answer 1

Dear student
$$\begin{gathered} \mathrm{First}\mathrm{row}\mathrm{has}30\mathrm{mangoes},\mathrm{second}\mathrm{row}\mathrm{has}27\mathrm{mangoes},\mathrm{third}\mathrm{row}\mathrm{has}24\mathrm{mangoes},... \\ \mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{mangoes}=162 \\ \mathrm{So},30,27,24,....\mathrm{is}\mathrm{forming}\mathrm{an}\mathrm{A}.\mathrm{P}. \\ \mathrm{with}{\mathrm{S}}_{\mathrm{n}}=162,\mathrm{a}=30,\mathrm{d}=-3 \\ \mathrm{So},{\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[2\mathrm{a}+(\mathrm{n}-1)\mathrm{d}\right] \\ \Rightarrow 162=\frac{\mathrm{n}}{2}\left[60+(\mathrm{n}-1)(-3)\right] \\ \Rightarrow 324=\mathrm{n}\left[60-3\mathrm{n}+3\right] \\ \Rightarrow 324=60\mathrm{n}-3{\mathrm{n}}^2+3\mathrm{n} \\ \Rightarrow 3{\mathrm{n}}^2-63\mathrm{n}+324=0 \\ \Rightarrow \mathrm{n}=\frac{63\pm \sqrt{3969-3888}}{6} \\ \Rightarrow \mathrm{n}=\frac{63\pm 9}{6} \\ \Rightarrow \mathrm{n}=12\mathrm{or}9 \\ \mathrm{So},\mathrm{there}\mathrm{are}12\mathrm{or}9\mathrm{rows}. \\ \mathrm{Mangoes}\mathrm{in}\mathrm{the}\mathrm{last}\mathrm{row}\mathrm{if}\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{are}12=30+(12-1)(-3)=-3(\mathrm{not}\mathrm{possible}) \\ \mathrm{Mangoes}\mathrm{in}\mathrm{the}\mathrm{last}\mathrm{row}\mathrm{if}\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{are}9=30+(9-1)(-3)=6 \\ \mathrm{So},\mathrm{number}\mathrm{of}\mathrm{rows}\mathrm{are}9 \end{gathered}$$
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