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Sequences and Series — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSequences and Series5 Marks
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

Here, first term $a = 5$ and the common difference $d = 2$ let the carpenter will take $n$ days to finish the job
$S_n = 192$
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(2\text{a}+(\text{n}-1)\text{d}]$
$192=\frac{\text{n}}{2}[2\times5+(\text{n}-1)2]$
$\Rightarrow 192 = n[10 + 2n - 2]$
$\Rightarrow 384 = n(2n + 8)$
$\Rightarrow 384 = 2n2 + 8n$
$\Rightarrow 2n^2 + 8n - 384 = 0$
$\Rightarrow n^2 + 4n - 192 = 0$
$\Rightarrow n^2 + 16n - 12n - 192 = 0$
$\Rightarrow n(n + 16) - 12(n + 16) = 0$
$\Rightarrow (n - 12)(n + 16) = 0$
$\Rightarrow n = 12\  [\because\ \text{n}\neq-16]$
Hence$,$ the required number of days $ = 12.$

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