If 7 times the seventh term of the AP is equal to 5 times the fifth term, then find the value of its `12^(th)` term.

Answer

To solve the problem, we need to find the value of the 12th term of an arithmetic progression (AP) given that 7 times the seventh term is equal to 5 times the fifth term.

**Step 1: Understand the terms of the AP**
The nth term of an AP can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

**Step 2: Write down the expressions for the 5th and 7th terms**
- The 5th term (\( a_5 \)) can be expressed as:
\[ a_5 = a + 4d \]
- The 7th term (\( a_7 \)) can be expressed as:
\[ a_7 = a + 6d \]

**Step 3: Set up the equation based on the given condition**
According to the problem, we have:
\[ 7 \times a_7 = 5 \times a_5 \]
Substituting the expressions for \( a_5 \) and \( a_7 \):
\[ 7(a + 6d) = 5(a + 4d) \]

**Step 4: Expand both sides of the equation**
Expanding the left side:
\[ 7a + 42d \]
Expanding the right side:
\[ 5a + 20d \]

So, we have:
\[ 7a + 42d = 5a + 20d \]

**Step 5: Rearrange the equation**
Now, let's rearrange the equation to isolate \( a \) and \( d \):
\[ 7a - 5a = 20d - 42d \]
This simplifies to:
\[ 2a = -22d \]

**Step 6: Solve for \( a \)**
Dividing both sides by 2 gives:
\[ a = -11d \]

**Step 7: Find the 12th term**
Now, we need to find the 12th term (\( a_{12} \)):
\[ a_{12} = a + 11d \]
Substituting the value of \( a \):
\[ a_{12} = -11d + 11d \]
This simplifies to:
\[ a_{12} = 0 \]

Thus, the value of the 12th term is:
\[ \boxed{0} \]

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