Four numbers are inserted between the numbers 5 and 95 such that an A.P. results. Find the biggest of these four numbers.

Options

A. 77
B. 85
C. 70
D. 80

Answer

To solve the problem of finding the biggest of the four numbers inserted between 5 and 95 such that they form an arithmetic progression (A.P.), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the terms of the A.P.**:
- The first term (a) is 5.
- The last term (a₆) is 95.
- We need to insert 4 numbers between them, which means we will have a total of 6 terms: 5, A₁, A₂, A₃, A₄, and 95.

2. **Use the formula for the nth term of an A.P.**:
- The nth term of an A.P. is given by the formula:
\[ T_n = a + (n - 1) \cdot d \]
- Here, \( T_6 = 95 \), \( a = 5 \), and \( n = 6 \).

3. **Set up the equation for the 6th term**:
\[ 95 = 5 + (6 - 1) \cdot d \]
\[ 95 = 5 + 5d \]

4. **Solve for d**:
- Rearranging the equation:
\[ 95 - 5 = 5d \]
\[ 90 = 5d \]
\[ d = \frac{90}{5} = 18 \]

5. **Find the 5th term (A₄)**:
- The 5th term (A₄) can be calculated using the formula:
\[ A₄ = a + 4d \]
- Substituting the values of a and d:
\[ A₄ = 5 + 4 \cdot 18 \]
\[ A₄ = 5 + 72 = 77 \]

6. **Conclusion**:
- The biggest of the four numbers inserted between 5 and 95 is **77**.