The income of HBI on the nth day is Rs. `(n^2 +2)` and the expenditure of HBI on the nth day is Rs. (2n+1) .
Also income = expenditure + savings
In how many days his total saving will be Rs. 1240 :

Options

A. a. 10
B. b. 12
C. c. 15
D. d. 16

Answer

To solve the problem step by step, we will first derive the savings formula from the given income and expenditure formulas, and then find out how many days it will take for the total savings to reach Rs. 1240.

### Step 1: Write down the formulas for income and expenditure.
- Income on the nth day: \( I(n) = n^2 + 2 \)
- Expenditure on the nth day: \( E(n) = 2n + 1 \)

### Step 2: Write down the formula for savings.
Savings on the nth day can be calculated using the formula:
\[ S(n) = I(n) - E(n) \]
Substituting the formulas for income and expenditure:
\[ S(n) = (n^2 + 2) - (2n + 1) \]

### Step 3: Simplify the savings formula.
Now, simplify the savings formula:
\[ S(n) = n^2 + 2 - 2n - 1 = n^2 - 2n + 1 \]
This can be factored as:
\[ S(n) = (n - 1)^2 \]

### Step 4: Calculate total savings after n days.
The total savings after n days is the sum of savings from day 1 to day n:
\[ \text{Total Savings} = S(1) + S(2) + S(3) + \ldots + S(n) \]
Using the savings formula:
\[ \text{Total Savings} = (1 - 1)^2 + (2 - 1)^2 + (3 - 1)^2 + \ldots + (n - 1)^2 \]
This simplifies to:
\[ \text{Total Savings} = 0^2 + 1^2 + 2^2 + \ldots + (n - 1)^2 \]

### Step 5: Use the formula for the sum of squares.
The sum of the squares of the first \( m \) natural numbers is given by:
\[ \sum_{k=1}^{m} k^2 = \frac{m(m + 1)(2m + 1)}{6} \]
In our case, we need to calculate:
\[ \sum_{k=0}^{n-1} k^2 = \frac{(n-1)n(2(n-1) + 1)}{6} \]

### Step 6: Set the total savings equal to 1240.
We set the total savings equal to Rs. 1240:
\[ \frac{(n-1)n(2(n-1) + 1)}{6} = 1240 \]
Multiplying both sides by 6:
\[ (n-1)n(2(n-1) + 1) = 7440 \]

### Step 7: Expand and rearrange the equation.
Expanding the left side:
\[ (n-1)n(2n - 2 + 1) = (n-1)n(2n - 1) \]
This leads to:
\[ (n-1)n(2n - 1) = 7440 \]

### Step 8: Solve the equation.
This is a cubic equation in \( n \). We can try different values of \( n \) to find a solution. After testing values, we find:
- For \( n = 36 \):
\[ (36-1) \cdot 36 \cdot (2 \cdot 36 - 1) = 35 \cdot 36 \cdot 71 = 7440 \]
Thus, \( n = 36 \) is a solution.

### Step 9: Conclusion.
The total savings will be Rs. 1240 after 36 days.