Sheela has a recurring deposit account in a bank of Rs 2,000 per month at the rate of 10% per annum. If she gets Rs 83,100 at the time of maturity, find the total time (in years) for which the account was held.

Answer

To solve the problem step-by-step, we will follow the process of calculating the maturity amount of a recurring deposit account.

### Step 1: Identify the given values
- Monthly deposit (P) = Rs 2000
- Rate of interest (r) = 10% per annum
- Maturity amount (M) = Rs 83,100

### Step 2: Calculate the total amount deposited
If Sheela deposits Rs 2000 every month for 'n' months, the total amount deposited (Principal) will be:
\[ \text{Total Principal} = P \times n = 2000 \times n \]

### Step 3: Calculate the Simple Interest (SI)
The formula for Simple Interest on a recurring deposit is:
\[ SI = \frac{P \times n(n + 1)}{2 \times 12} \times \frac{r}{100} \]
Substituting the values:
\[ SI = \frac{2000 \times n(n + 1)}{2 \times 12} \times \frac{10}{100} \]
\[ SI = \frac{2000 \times n(n + 1)}{24} \times \frac{1}{10} \]
\[ SI = \frac{2000 \times n(n + 1)}{240} \]
\[ SI = \frac{25 \times n(n + 1)}{3} \]

### Step 4: Write the equation for maturity amount
The maturity amount (M) can be expressed as:
\[ M = \text{Total Principal} + SI \]
Substituting the values:
\[ 83100 = 2000n + \frac{25n(n + 1)}{3} \]

### Step 5: Clear the equation
Multiply the entire equation by 3 to eliminate the fraction:
\[ 3 \times 83100 = 3 \times 2000n + 25n(n + 1) \]
\[ 249300 = 6000n + 25n^2 + 25n \]
\[ 249300 = 25n^2 + 6025n \]

### Step 6: Rearrange to form a quadratic equation
Rearranging gives:
\[ 25n^2 + 6025n - 249300 = 0 \]

### Step 7: Simplify the equation
Divide the entire equation by 25:
\[ n^2 + 241n - 9972 = 0 \]

### Step 8: Factor the quadratic equation
To factor the quadratic equation, we look for two numbers that multiply to -9972 and add to 241. The factors are:
\[ (n - 36)(n + 277) = 0 \]

### Step 9: Solve for n
Setting each factor to zero gives:
1. \( n - 36 = 0 \) → \( n = 36 \)
2. \( n + 277 = 0 \) → \( n = -277 \) (not valid since time cannot be negative)

Thus, \( n = 36 \) months.

### Step 10: Convert months to years
To find the time in years:
\[ \text{Time in years} = \frac{n}{12} = \frac{36}{12} = 3 \text{ years} \]

### Final Answer
The total time for which the account was held is **3 years**.

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