Only two entries are known of the following Arithmetic progression :
__, 5 ___ , ___ , 14 , __ , ___
What should be the number just after 14 ?

Options

A. 17
B. 18
C. 19
D. 20

Answer

To solve the problem of finding the number just after 14 in the given arithmetic progression (AP), we can follow these steps:

### Step 1: Identify the known terms
We have the following terms in the AP:
- The second term (A2) is 5.
- The fourth term (A4) is 14.

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

### Step 3: Set up equations based on the known terms
From the second term:
\[ A_2 = A + D = 5 \quad \text{(1)} \]

From the fourth term:
\[ A_4 = A + 3D = 14 \quad \text{(2)} \]

### Step 4: Solve the equations
Now, we can solve these two equations.

Subtract equation (1) from equation (2):
\[ (A + 3D) - (A + D) = 14 - 5 \]
This simplifies to:
\[ 2D = 9 \]
Thus, we find:
\[ D = \frac{9}{2} = 4.5 \]

### Step 5: Substitute D back into one of the equations
Now, substitute \( D \) back into equation (1) to find \( A \):
\[ A + 4.5 = 5 \]
This gives:
\[ A = 5 - 4.5 = 0.5 \]

### Step 6: Find the sixth term (A6)
Now that we have \( A \) and \( D \), we can find the sixth term (A6):
\[ A_6 = A + 5D \]
Substituting the values we found:
\[ A_6 = 0.5 + 5 \times 4.5 \]
Calculating this:
\[ A_6 = 0.5 + 22.5 = 23 \]

### Conclusion
The number just after 14 in the arithmetic progression is **23**.

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