If 8 times the eighth term of an A.P. is equal to 15 times its fifteenth term, find its 23rd term.

Answer

To solve the problem step by step, we will use the formula for the \( n \)-th term of an arithmetic progression (A.P.), which is given by:

\[ T_n = a + (n - 1)d \]

where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

### Step 1: Write the equations for the 8th and 15th terms
According to the problem, we have:

\[ 8 \times T_8 = 15 \times T_{15} \]

Using the formula for the \( n \)-th term:

\[ T_8 = a + (8 - 1)d = a + 7d \]
\[ T_{15} = a + (15 - 1)d = a + 14d \]

Substituting these into the equation gives:

\[ 8(a + 7d) = 15(a + 14d) \]

### Step 2: Expand both sides
Expanding both sides of the equation:

\[ 8a + 56d = 15a + 210d \]

### Step 3: Rearrange the equation
Now, we will rearrange the equation to bring all terms involving \( a \) and \( d \) to one side:

\[ 8a + 56d - 15a - 210d = 0 \]

This simplifies to:

\[ -7a - 154d = 0 \]

### Step 4: Factor out common terms
Factoring out \(-7\):

\[ 7a + 154d = 0 \]

Dividing through by 7 gives:

\[ a + 22d = 0 \]

### Step 5: Solve for \( a \)
From the equation \( a + 22d = 0 \), we can express \( a \) in terms of \( d \):

\[ a = -22d \]

### Step 6: Find the 23rd term
Now we need to find the 23rd term \( T_{23} \):

\[ T_{23} = a + (23 - 1)d = a + 22d \]

Substituting \( a = -22d \) into the equation:

\[ T_{23} = -22d + 22d = 0 \]

### Final Answer
Thus, the 23rd term \( T_{23} \) is:

\[ \boxed{0} \]