To solve the problem of determining the total number of installments, let's summarize the given information:

1. Loan Amount: ₹1,39,000
2. Initial Installment: ₹13,000
3. Installment Increases by ₹1300 every month.

This forms an arithmetic series where:

• The first installment $a = 13,000$
• The common difference $d = 1,300$
• The total sum $S_n = 1,39,000$

We need to find the number of installments $n$. The sum of an arithmetic series is given by the formula:

$S_n = \frac{n}{2} \times (2a + (n-1) \times d)$

Substituting the known values:

$1,39,000 = \frac{n}{2} \times (2 \times 13,000 + (n-1) \times 1,300)$

Now, I'll solve this equation to find $n$.The solution for $n$ from the equation includes two values:

1. $n = \frac{-19}{2} + \frac{3\sqrt{22841}}{26}$
2. $n = \frac{-3\sqrt{22841}}{26} - \frac{19}{2}$

We are only interested in the positive value of $n$. Let's simplify and approximate this expression to find the actual number of installments.The positive solution for the number of installments $n$ is approximately 7.94, which rounds to 8 installments.

Thus, your father would have to pay a total of 8 installments to repay the loan of ₹1,39,000.

Would you like more details or have any questions?

Here are some related questions:

1. How does the total repayment change if the monthly increase is larger?
2. How do we calculate the sum of installments if they increase non-linearly?
3. Can the installment pattern be modeled using a geometric series?
4. What would the total sum be if the loan had no monthly increase?
5. How can we calculate the effective interest rate from this pattern?

Tip: For any loan or debt repayment involving installment increases, using an arithmetic progression helps break down the monthly costs easily.