Question:
A carpenter was hired to build 192 window frames. The first day he made 5 frames and each day, thereafter he made 2 more frames than he made the day before. How many days did he take to finish the job?
Solution:
Let the carpenter take n days to finish the job.
To Find: n
He builds 5 frames on day 1, 7 on day 2, 9 on day 3 and so on.
So it forms an AP 5, 7, 9, 11,… and so on.
We need to find the number of terms in this AP such that the sum of the AP will be equal to 192
Given: Sum of AP = 192
$\frac{n}{2}[10+(n-1) 2]=192$
$\Rightarrow \frac{n}{2}[n+8]=192 \Rightarrow n(n+8)=192 \times 2=16 \times 24$
$\Rightarrow \mathrm{n}=16$
He finishes the job in 16 days.