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Find the total  number of installments he has to pay 

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.

Your father wants to buy a plot and plans to take a loan from a bank for that. He repays the total loan of ₹1,39,000 by paying every month starting with the first installment of ₹13,000. Also, he increases the installment by ₹1,300 every month. Find the total number of installments he has to pay.

Learn how to calculate the total number of installments to repay a loan of ₹1,39,000 using arithmetic progression. The solution includes key formulas like the sum of an arithmetic series and considers monthly installment increases.

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