(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

Let's solve the problem step by step.

### Part (i)

**Step 1: Set up the equation based on the given condition.**

We are given that 10 times the 10th term is equal to 15 times the 15th term of an arithmetic progression (A.P.).

Let the first term of the A.P. be \( A \) and the common difference be \( D \).

The 10th term \( A_{10} \) can be expressed as:
\[ A_{10} = A + 9D \]
The 15th term \( A_{15} \) can be expressed as:
\[ A_{15} = A + 14D \]

According to the problem:
\[ 10 \cdot A_{10} = 15 \cdot A_{15} \]
Substituting the expressions for \( A_{10} \) and \( A_{15} \):
\[ 10(A + 9D) = 15(A + 14D) \]

**Step 2: Simplify the equation.**

Expanding both sides:
\[ 10A + 90D = 15A + 210D \]

**Step 3: Rearrange the equation.**

Rearranging gives:
\[ 10A - 15A + 90D - 210D = 0 \]
\[ -5A - 120D = 0 \]

**Step 4: Solve for \( A \).**

From the above equation:
\[ -5A = 120D \implies A = -24D \]

**Step 5: Find the 25th term of the A.P.**

The 25th term \( A_{25} \) can be expressed as:
\[ A_{25} = A + 24D \]
Substituting \( A = -24D \):
\[ A_{25} = -24D + 24D = 0 \]

Thus, the 25th term of the A.P. is:
\[ \boxed{0} \]

### Part (ii)

**Step 1: Set up the equation based on the given condition.**

We are given that 17 times the 17th term is equal to 18 times the 18th term of an A.P.

Using the same notation as before:
\[ A_{17} = A + 16D \]
\[ A_{18} = A + 17D \]

According to the problem:
\[ 17 \cdot A_{17} = 18 \cdot A_{18} \]
Substituting the expressions for \( A_{17} \) and \( A_{18} \):
\[ 17(A + 16D) = 18(A + 17D) \]

**Step 2: Simplify the equation.**

Expanding both sides:
\[ 17A + 272D = 18A + 306D \]

**Step 3: Rearrange the equation.**

Rearranging gives:
\[ 17A - 18A + 272D - 306D = 0 \]
\[ -A - 34D = 0 \]

**Step 4: Solve for \( A \).**

From the above equation:
\[ -A = 34D \implies A = -34D \]

**Step 5: Find the 35th term of the A.P.**

The 35th term \( A_{35} \) can be expressed as:
\[ A_{35} = A + 34D \]
Substituting \( A = -34D \):
\[ A_{35} = -34D + 34D = 0 \]

Thus, the 35th term of the A.P. is:
\[ \boxed{0} \]