To solve this problem, we first calculate the efficiency of each person and then determine how the cumulative work done per day increases with each new person joining.

Step 1: Calculate Individual Efficiency

If a person can complete the entire job in 120 days, his work rate is: $\frac{1}{120} \text{ of the job per day}$

Step 2: Cumulative Work Done Per Day

• On Day 1, only one person works, so the work done is: $\frac{1}{120}$
• On Day 2, two people work, so the work done is: $2 \times \frac{1}{120} = \frac{2}{120} = \frac{1}{60}$
• On Day 3, three people work, so the work done is: $3 \times \frac{1}{120} = \frac{3}{120} = \frac{1}{40}$
• Similarly, on Day $n$, $n$ people are working, and the work done on that day is: $\frac{n}{120}$

Step 3: Set Up the Equation to Find the Total Days Required

Let the total number of days required to complete the job be $D$. The sum of work done over these $D$ days must equal 1 (the whole job).

So, we have: $\frac{1}{120} + \frac{2}{120} + \frac{3}{120} + \cdots + \frac{D}{120} = 1$

This can be factored as: $\frac{1}{120} (1 + 2 + 3 + \cdots + D) = 1$

Step 4: Sum of the First $D$ Natural Numbers

The sum of the first $D$ natural numbers is given by: $1 + 2 + 3 + \cdots + D = \frac{D(D + 1)}{2}$

Thus, the equation becomes: $\frac{1}{120} \times \frac{D(D + 1)}{2} = 1$

Multiplying both sides by 120 to clear the fraction: $\frac{D(D + 1)}{2} = 120$

Multiply by 2 to simplify: $D(D + 1) = 240$

Step 5: Solve the Quadratic Equation

We now have: $D^2 + D - 240 = 0$

Using the quadratic formula: $D = \frac{-1 \pm \sqrt{1 + 4 \times 240}}{2}$ $D = \frac{-1 \pm \sqrt{961}}{2}$ $D = \frac{-1 \pm 31}{2}$

This gives: $D = \frac{30}{2} = 15 \quad \text{or} \quad D = \frac{-32}{2} = -16$

Since $D$ must be positive, we have: $D = 15$

Conclusion

The job will be completed in 15 days.

Would you like further clarification on any step?

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Here are five related questions that expand upon this topic:

1. How would the solution change if each person could complete the job in 100 days instead of 120 days?
2. What if a new person joined every two days instead of every day?
3. How does the work rate change if each person joining is only half as efficient as the previous person?
4. What if there were a cap on the number of people allowed to join the work?
5. How would the equation adjust if the first person works alone for multiple days before others join?

Tip: Breaking down complex word problems into smaller steps, such as identifying individual rates and cumulative work, makes them easier to solve.