For an AP with first term u and common difference v. the `p^(th)` term is 15 uv more than the `q^(th)` term. Which one of the following is correct ?

Options

A. p=q+15v
B. p=q+15u
C. p=q+14v
D. p=q+14v

Answer

To solve the problem step-by-step, we will analyze the information given about the arithmetic progression (AP) and derive the relationship between the terms.

### Step 1: Understand the terms of the AP
The first term of the AP is given as \( u \) and the common difference is \( v \).

### Step 2: Write the formula for the \( p^{th} \) and \( q^{th} \) terms
The \( n^{th} \) term of an AP can be expressed as:
\[ T_n = a + (n - 1)d \]
For our case:
- The \( p^{th} \) term (\( T_p \)) is:
\[ T_p = u + (p - 1)v \]
- The \( q^{th} \) term (\( T_q \)) is:
\[ T_q = u + (q - 1)v \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the \( p^{th} \) term is 15uv more than the \( q^{th} \) term. Therefore, we can write:
\[ T_p = T_q + 15uv \]
Substituting the expressions for \( T_p \) and \( T_q \):
\[ u + (p - 1)v = (u + (q - 1)v) + 15uv \]

### Step 4: Simplify the equation
Now, we simplify the equation:
1. Start by canceling \( u \) from both sides:
\[ (p - 1)v = (q - 1)v + 15uv \]
2. Rearranging gives:
\[ pv - v = qv - v + 15uv \]
3. Further simplifying:
\[ pv - qv = 15uv \]
4. Factor out \( v \):
\[ v(p - q) = 15uv \]

### Step 5: Divide both sides by \( v \) (assuming \( v \neq 0 \))
\[ p - q = 15u \]
Thus, we can express \( p \) in terms of \( q \):
\[ p = q + 15u \]

### Conclusion
The correct relationship derived from the given information is:
\[ p = q + 15u \]

### Final Answer
The correct option is: **P is equal to Q plus 15U**.