If the ratio of the `11^(th)` term of an AP to its `18^(th)` term is `2 : 3`, find the ratio of the sum of the first five terms to the sum of its first 10 terms.

Answer

To solve the problem step by step, we will follow the given information and use the formulas related to Arithmetic Progressions (AP).

### Step 1: Understand the given information
We know that the ratio of the 11th term (A11) to the 18th term (A18) of an AP is given as 2:3.

### Step 2: Write the formula for the nth term of an AP
The nth term of an AP can be expressed as:
\[ A_n = A + (n-1) \cdot d \]
where \( A \) is the first term and \( d \) is the common difference.

### Step 3: Express A11 and A18 using the formula
Using the formula for the nth term:
- The 11th term (A11) is:
\[ A_{11} = A + 10d \]
- The 18th term (A18) is:
\[ A_{18} = A + 17d \]

### Step 4: Set up the ratio based on the given information
We can set up the equation based on the ratio:
\[ \frac{A_{11}}{A_{18}} = \frac{2}{3} \]
Substituting the expressions for A11 and A18:
\[ \frac{A + 10d}{A + 17d} = \frac{2}{3} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3(A + 10d) = 2(A + 17d) \]

### Step 6: Expand and simplify the equation
Expanding both sides:
\[ 3A + 30d = 2A + 34d \]

Now, rearranging the terms:
\[ 3A - 2A = 34d - 30d \]
This simplifies to:
\[ A = 4d \]

### Step 7: Find the sums S5 and S10
The sum of the first n terms of an AP is given by:
\[ S_n = \frac{n}{2} \cdot (2A + (n-1)d) \]

#### For S5:
Using \( n = 5 \):
\[ S_5 = \frac{5}{2} \cdot (2A + 4d) \]
Substituting \( A = 4d \):
\[ S_5 = \frac{5}{2} \cdot (2(4d) + 4d) = \frac{5}{2} \cdot (8d + 4d) = \frac{5}{2} \cdot 12d = 30d \]

#### For S10:
Using \( n = 10 \):
\[ S_{10} = \frac{10}{2} \cdot (2A + 9d) \]
Substituting \( A = 4d \):
\[ S_{10} = 5 \cdot (2(4d) + 9d) = 5 \cdot (8d + 9d) = 5 \cdot 17d = 85d \]

### Step 8: Find the ratio of S5 to S10
Now, we can find the ratio:
\[ \frac{S_5}{S_{10}} = \frac{30d}{85d} = \frac{30}{85} = \frac{6}{17} \]

### Final Answer:
The ratio of the sum of the first 5 terms to the sum of the first 10 terms is:
\[ \frac{6}{17} \]

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