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A carpenter was hired to build 192 window frames. The first day he made five farmes 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?
Solution:
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Verified Answer
Here, $a=5$ and $d=2$
Let he finished the job in $n$ days.
Then, $S_n=192$
$$
\begin{aligned}
&S_n=\frac{n}{2}[2 a+(n-1) d] \\
&\Rightarrow 192=\frac{n}{2}[2 \times 5+(n-1) 2] \\
&\Rightarrow 192=\frac{n}{2}[10+2 n-2] \\
&\Rightarrow 192=\frac{n}{2}[8+2 n] \Rightarrow 192=4 n+n^2 \\
&\Rightarrow n^2+4 n-192=0 \Rightarrow(n-12)(n+16)=0 \\
&\Rightarrow n=12,-16 \quad \quad[\because n \neq-16] \\
&\therefore n=12 \quad[\because n
\end{aligned}
$$
Let he finished the job in $n$ days.
Then, $S_n=192$
$$
\begin{aligned}
&S_n=\frac{n}{2}[2 a+(n-1) d] \\
&\Rightarrow 192=\frac{n}{2}[2 \times 5+(n-1) 2] \\
&\Rightarrow 192=\frac{n}{2}[10+2 n-2] \\
&\Rightarrow 192=\frac{n}{2}[8+2 n] \Rightarrow 192=4 n+n^2 \\
&\Rightarrow n^2+4 n-192=0 \Rightarrow(n-12)(n+16)=0 \\
&\Rightarrow n=12,-16 \quad \quad[\because n \neq-16] \\
&\therefore n=12 \quad[\because n
\end{aligned}
$$
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