Let's break down the question based on the provided image.

The question states that:

• Your father wants to buy a flat and plans to take a loan from a bank.
• The total loan amount is Rs. 15,39,000.
• He will repay the loan in installments, starting with Rs. 13,000 in the first month.
• He increases the installment by Rs. 1,300 every month.

This forms an arithmetic progression (AP) where:

• The first installment $a_1 = Rs. 13,000$,
• The common difference $d = Rs. 1,300$,
• The total loan amount $S = Rs. 15,39,000$.

In an arithmetic progression, the sum of $n$ terms is given by:

$S_n = \frac{n}{2} [2a_1 + (n-1)d]$

Where:

• $n$ is the number of terms (installments),
• $S_n$ is the total sum (loan amount),
• $a_1$ is the first term (installment),
• $d$ is the common difference.

We need to find $n$, the number of installments. Let's solve for $n$ using the values provided.

$S_n = \frac{n}{2} [2 \times 13000 + (n - 1) \times 1300]$ $1539000 = \frac{n}{2} [26000 + 1300(n - 1)]$

Now I'll calculate the value of $n$.The values of $n$ (the number of installments) obtained include negative solutions, which are not physically meaningful for this context. Therefore, only the positive solution is valid. After calculation, the total number of installments turns out to be approximately 22.66.

Since the number of installments must be a whole number, he will need to pay 23 installments to fully repay the loan.

Would you like more details or have any other questions?

Here are some related questions:

1. How would the total change if the initial installment was different?
2. What is the total amount paid by the last installment?
3. How does increasing the monthly installment affect the total number of payments?
4. What if the increase per month was Rs. 1,000 instead of Rs. 1,300?
5. How can you calculate the loan if the bank has interest?

Tip: The formula for an arithmetic progression can also be used to calculate cumulative payments over time.