In a state, all the school teachers decided to help those underpriveleged children in the streets by donating some money every year to the child welfare fund. In the first year, they donated Rs. 10,000 in the second year they donated Rs. 20,000 and in the third year they donated Rs. 30,000 and they continue to pay for 10 years. Find the total amount that will be doneated after 10 years.

Answer

To find the total amount donated by the school teachers over 10 years, we can observe that the donations form an arithmetic progression (AP). Let's break down the solution step by step.

### Step 1: Identify the first term and common difference
The first term (a) of the donation series is Rs. 10,000 (the amount donated in the first year). The common difference (d) is the difference between the second and first term, which is:
\[ d = 20,000 - 10,000 = 10,000 \]

### Step 2: Determine the number of terms
The number of terms (n) is 10, as the donations are made for 10 years.

### Step 3: Use the formula for the sum of an arithmetic series
The formula for the sum of the first n terms (S_n) of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1) \cdot d) \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
- \( n = 10 \)
- \( a = 10,000 \)
- \( d = 10,000 \)

So,
\[ S_{10} = \frac{10}{2} \times (2 \cdot 10,000 + (10 - 1) \cdot 10,000) \]

### Step 5: Simplify the expression
Calculating step by step:
1. Calculate \( \frac{10}{2} = 5 \)
2. Calculate \( 2 \cdot 10,000 = 20,000 \)
3. Calculate \( (10 - 1) \cdot 10,000 = 9 \cdot 10,000 = 90,000 \)
4. Now, combine these: \( 20,000 + 90,000 = 110,000 \)

Putting it all together:
\[ S_{10} = 5 \times 110,000 = 550,000 \]

### Final Answer
The total amount donated after 10 years is Rs. 550,000.

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