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 find the number just after 14 in the given arithmetic progression (AP) `?, 5, ?, ?, 14, ?`, we can follow these steps:

### Step 1: Identify the terms in the AP
In an arithmetic progression, the terms can be expressed using the first term \( a \) and the common difference \( d \). The terms can be represented as:
- 1st term: \( a \)
- 2nd term: \( a + d \)
- 3rd term: \( a + 2d \)
- 4th term: \( a + 3d \)
- 5th term: \( a + 4d \)
- 6th term: \( a + 5d \)

From the question, we know:
- The 2nd term \( a + d = 5 \)
- The 5th term \( a + 4d = 14 \)

### Step 2: Set up the equations
From the known terms, we can set up the following equations:
1. \( a + d = 5 \) (Equation 1)
2. \( a + 4d = 14 \) (Equation 2)

### Step 3: Solve the equations
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 4d) - (a + d) = 14 - 5 \]
This simplifies to:
\[ 3d = 9 \]
Now, divide both sides by 3:
\[ d = 3 \]

### Step 4: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 3 = 5 \]
Subtract 3 from both sides:
\[ a = 2 \]

### Step 5: Find the 6th term
Now we can find the 6th term using the formula \( a + 5d \):
\[ 6th \ term = a + 5d = 2 + 5 \times 3 \]
Calculating this gives:
\[ = 2 + 15 = 17 \]

### Final Answer
The number just after 14 in the arithmetic progression is **17**.

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