Bank Deposit Growth with Daily Compounding

The problem describes a bank deposit where interest is compounded daily. This means the growth of the deposit follows the principles of compound interest. We are given the value of the deposit at two different points in time (500 days and 1000 days) and asked to find its value at a third point (1500 days).

Let the initial deposit amount be \(P\). Let the value of the deposit after \(t\) days be \(A(t)\). Since the interest is compounded daily, the growth over any fixed period is proportional. Let's consider the 500-day interval as a single period for simplicity in understanding the growth pattern.

We are given:

• Value after 500 days, \(A(500) = 80,000\) Rs.
• Value after 1000 days, \(A(1000) = 88,000\) Rs.

We need to find the value after 1500 days, \(A(1500)\).

In compound interest, the amount grows by a constant factor over equal periods. Let the growth factor for a period of 500 days be \(G\).

So, the value after 500 days is the initial principal multiplied by the growth factor for 500 days:

\(A(500) = P \times G = 80,000\) (Equation 1)

The value after 1000 days is the initial principal multiplied by the growth factor for two periods of 500 days (i.e., \(G\) applied twice):

\(A(1000) = P \times G^2 = 88,000\) (Equation 2)

Now, we can find the growth factor \(G\) by dividing Equation 2 by Equation 1:

\(\frac{A(1000)}{A(500)} = \frac{P \times G^2}{P \times G} = \frac{88,000}{80,000}\)

\(G = \frac{88}{80} = \frac{11}{10} = 1.1\)

The growth factor for every 500-day period is 1.1.

Calculating Value after 1500 Days

The value after 1500 days is the initial principal multiplied by the growth factor for three periods of 500 days (i.e., \(G\) applied three times):

\(A(1500) = P \times G^3\)

We can also think of the value at 1500 days as the value at 1000 days multiplied by the growth factor \(G\) for the next 500 days (from day 1000 to day 1500).

\(A(1500) = A(1000) \times G\)

Using the values we know:

\(A(1500) = 88,000 \times 1.1\)

\(A(1500) = 88,000 \times \frac{11}{10}\)

\(A(1500) = 8800 \times 11\)

Performing the multiplication:

\(8800 \times 11 = 8800 \times (10 + 1) = 88000 + 8800 = 96800\)

So, the value of the deposit after 1500 days would be 96,800 Rs.

Summary of Growth

Let's look at the values at each 500-day mark:

The deposit grows from 80,000 to 88,000 over the second 500-day period, which is a growth of 8,000. Over the third 500-day period, it grows from 88,000 by the same factor of 1.1.

Growth in 1st 500 days (P to 80000)

Growth in 2nd 500 days (80000 to 88000): \(88000 - 80000 = 8000\)

Growth in 3rd 500 days (88000 to 96800): \(96800 - 88000 = 8800\)

The growth amount also increases with each period due to compounding.

The value of the deposit after 1500 days is 96,800 Rs.