Two APs have the same common difference. The first term of one of these is -1 and that of the other is -8. The difference between their 4th terms is

Options

A. -1
B. -8
C. 7
D. -9

Answer

To solve the problem step by step, we will calculate the difference between the fourth terms of the two arithmetic progressions (APs).

### Step 1: Identify the first terms and common difference
Let:
- The first term of the first AP (A1) = -1
- The first term of the second AP (A2) = -8
- The common difference (D) for both APs = D (same for both)

### Step 2: Write the formula for the nth term of an AP
The nth term of an arithmetic progression can be calculated using the formula:
\[ A_n = A + (n - 1)D \]
where:
- \( A \) is the first term,
- \( n \) is the term number,
- \( D \) is the common difference.

### Step 3: Calculate the fourth term of the first AP
Using the formula for the first AP:
\[ A_4 = A_1 + (4 - 1)D \]
Substituting the values:
\[ A_4 = -1 + 3D \]

### Step 4: Calculate the fourth term of the second AP
Using the formula for the second AP:
\[ A'_4 = A_2 + (4 - 1)D \]
Substituting the values:
\[ A'_4 = -8 + 3D \]

### Step 5: Find the difference between the fourth terms
Now, we need to find the difference between the fourth terms of the two APs:
\[ \text{Difference} = A_4 - A'_4 \]
Substituting the expressions we found:
\[ \text{Difference} = (-1 + 3D) - (-8 + 3D) \]

### Step 6: Simplify the expression
Now, simplifying the expression:
\[ \text{Difference} = -1 + 3D + 8 - 3D \]
The \( 3D \) terms cancel each other out:
\[ \text{Difference} = -1 + 8 \]
\[ \text{Difference} = 7 \]

### Conclusion
The difference between the fourth terms of the two APs is **7**.

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