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A person can complete a job in $120$ days. He works alone on Day $1$. On Day $2$, he is joined by another person who also can complete the job in exactly $120$ days. On Day $3$, they are joined by another person of equal efficiency. Like this, everyday a new person with the same efficiency joins the work. How many days are required to complete the job?

1. $15$
2. $35$
3. $23$
4. $40$

Given that, $1\;\text{person} = 120\;\text{days} \Rightarrow 120\;\text{persons} = 1\;\text{day}$

Let, the number of days are required to complete the job be $n$.

$1+2+3+4+5+ \ldots + n = 120 \quad [\because \text{Efficiency are equal}]$

$\Rightarrow \frac{n(n+1)}{2} = 120$

$\Rightarrow n^{2} + n = 240$

$\Rightarrow n^{2} + n – 240 = 0$

$\Rightarrow n^{2} + 16n – 15n – 240 = 0$

$\Rightarrow n(n+16) -15(n+16) = 0$

$\Rightarrow (n+16)(n-15) = 0$

$\Rightarrow n = 15, n = -16\;(\text{The number of days can’t be negative})$

$\Rightarrow \boxed{n = 15}$

$\therefore$ The number of days is required to complete the job $= 15$ days.

Correct Answer $: \text{A}$
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