A sum of 8000 is invested at 10% simple interest per annum. Calculate the interest at the end of each year.
Does the sequence of interests at the end of consecutive years form an A.P.? If yes write its first term and the common difference.

Answer

To solve the problem step by step, we will calculate the simple interest for each year and determine if the sequence of interests forms an arithmetic progression (A.P.).

### Step 1: Identify the given values
- Principal (P) = 8000
- Rate of interest (R) = 10% per annum

### Step 2: Write the formula for Simple Interest (SI)
The formula for calculating Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest
- T = Time in years

### Step 3: Calculate the Simple Interest for each year
1. **Interest at the end of the first year (T = 1):**
\[ \text{SI}_1 = \frac{8000 \times 10 \times 1}{100} = \frac{80000}{100} = 800 \]

2. **Interest at the end of the second year (T = 2):**
\[ \text{SI}_2 = \frac{8000 \times 10 \times 2}{100} = \frac{160000}{100} = 1600 \]

3. **Interest at the end of the third year (T = 3):**
\[ \text{SI}_3 = \frac{8000 \times 10 \times 3}{100} = \frac{240000}{100} = 2400 \]

### Step 4: List the sequence of interests
- Interest at the end of the first year: 800
- Interest at the end of the second year: 1600
- Interest at the end of the third year: 2400

Thus, the sequence of interests is: 800, 1600, 2400.

### Step 5: Check if the sequence forms an A.P.
To determine if the sequence is an A.P., we need to check if the difference between consecutive terms is constant.

- Difference between the second and first term:
\[ 1600 - 800 = 800 \]

- Difference between the third and second term:
\[ 2400 - 1600 = 800 \]

Since the difference is the same (800), the sequence does form an A.P.

### Step 6: Identify the first term and the common difference
- First term (a) = 800
- Common difference (d) = 800

### Final Answer
The interest at the end of each year is:
- Year 1: 800
- Year 2: 1600
- Year 3: 2400

The sequence of interests does form an A.P. with:
- First term = 800
- Common difference = 800