A carpenter was hired to build 192 window frames. The first day h… - Vidyadip

Question

A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?

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Answer

Suppose carpenter took n days to finish his job. First day carpenter made five frames $\text{a}_1=5$ Each day atter first day he made two more frames $\text{d}=2$ $\therefore$ On $n^{th}$ day frames made by carpenter are, $\text{a}_\text{n}=\text{a}_\text{n}+(\text{n}-1)(\text{d})$ $\Rightarrow\text{a}_\text{n}5+(\text{n}-1)2$ Sum of all the frames till $n^{Ln}$ day is $\text{S}=\frac{\text{n}}{2}[\text{a}_1+\text{a}_\text{n}]$ $192=\frac{\text{n}}{2}[5+5+(\text{n}-1)2]$ $192=5\text{n}+\text{n}^2-\text{n}$ $\text{n}^2+4\text{n}-192=0$ $(\text{n}+16)(\text{n}-12)=0$ $\text{n}=-16$ or $\text{n}=12$ But number of days cannot be negative hence $\text{n}=12$ The carpenter took 12 days to finish his job.

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