(i) 10 times the 10th term and 15 times the 15th term of an A.P. are equal. Find the 25th term of this A.P .
(ii) 17 times the 17th term of an A.P. is equal to 18 times the 18th term. Find the 35th term of this progression.

Explanation

To solve the given problems step by step, let's break them down into parts.

### Part (i)

**Given:**
10 times the 10th term and 15 times the 15th term of an A.P. are equal.
We need to find the 25th term of this A.P.

**Step 1:** Write the expressions for the 10th and 15th terms of the A.P.
The nth term of an A.P. is given by the formula:
\[ A_n = A + (n-1)D \]
Where \( A \) is the first term and \( D \) is the common difference.

Thus,
- The 10th term \( A_{10} = A + 9D \)
- The 15th term \( A_{15} = A + 14D \)

**Step 2:** Set up the equation based on the problem statement.
According to the problem,
\[ 10 \cdot A_{10} = 15 \cdot A_{15} \]
Substituting the expressions we found:
\[ 10(A + 9D) = 15(A + 14D) \]

**Step 3:** Expand and simplify the equation.
Expanding both sides:
\[ 10A + 90D = 15A + 210D \]

**Step 4:** Rearrange the equation to isolate terms involving \( A \) and \( D \).
Bringing all terms involving \( A \) to one side and terms involving \( D \) to the other:
\[ 10A - 15A = 210D - 90D \]
This simplifies to:
\[ -5A = 120D \]
Thus,
\[ A = -\frac{120D}{5} = -24D \]

**Step 5:** Find the 25th term \( A_{25} \).
Using the formula for the 25th term:
\[ A_{25} = A + 24D \]
Substituting \( A = -24D \):
\[ A_{25} = -24D + 24D = 0 \]

**Conclusion:**
The 25th term of the A.P. is **0**.

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### Part (ii)

**Given:**
17 times the 17th term of an A.P. is equal to 18 times the 18th term.
We need to find the 35th term of this A.P.

**Step 1:** Write the expressions for the 17th and 18th terms of the A.P.
Using the same formula:
- The 17th term \( A_{17} = A + 16D \)
- The 18th term \( A_{18} = A + 17D \)

**Step 2:** Set up the equation based on the problem statement.
According to the problem,
\[ 17 \cdot A_{17} = 18 \cdot A_{18} \]
Substituting the expressions we found:
\[ 17(A + 16D) = 18(A + 17D) \]

**Step 3:** Expand and simplify the equation.
Expanding both sides:
\[ 17A + 272D = 18A + 306D \]

**Step 4:** Rearrange the equation to isolate terms involving \( A \) and \( D \).
Bringing all terms involving \( A \) to one side and terms involving \( D \) to the other:
\[ 17A - 18A = 306D - 272D \]
This simplifies to:
\[ -A = 34D \]
Thus,
\[ A = -34D \]

**Step 5:** Find the 35th term \( A_{35} \).
Using the formula for the 35th term:
\[ A_{35} = A + 34D \]
Substituting \( A = -34D \):
\[ A_{35} = -34D + 34D = 0 \]

**Conclusion:**
The 35th term of the A.P. is **0**.

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