Let `s_(1), s_(2), s_(3).... and t_(1), t_(2), t_(3)`.... are two arithmetic sequences such that `s_(1) = t_(1) != 0, s_(2) = 2t_(2) and sum_(i=1)^(10) s_(i) = sum_(i=1)^(15) t_(i)`. Then the value of `(s_(2) -s_(1))/(t_(2) - t_(1))` is

Options

A. `8//3`
B. `3//2`
C. 19/8
D. 2

Answer

To solve the problem, we need to analyze the given arithmetic sequences and use the properties of arithmetic progressions. Let's break it down step by step.

### Step 1: Define the sequences
Let:
- The first arithmetic sequence be \( s_n \) with first term \( s_1 = a_1 \) and common difference \( d_1 \).
- The second arithmetic sequence be \( t_n \) with first term \( t_1 = a_2 \) and common difference \( d_2 \).

### Step 2: Write the terms based on the definitions
From the definitions of the sequences, we have:
- \( s_1 = a_1 \)
- \( s_2 = a_1 + d_1 \)
- \( t_1 = a_2 \)
- \( t_2 = a_2 + d_2 \)

### Step 3: Use the given conditions
We know from the problem statement:
1. \( s_1 = t_1 \) implies \( a_1 = a_2 \).
2. \( s_2 = 2t_2 \) implies:
\[ a_1 + d_1 = 2(a_2 + d_2) \]
Substituting \( a_1 = a_2 \) gives:
\[ a_1 + d_1 = 2(a_1 + d_2) \]
Rearranging this:
\[ d_1 = 2a_1 + 2d_2 - a_1 = a_1 + 2d_2 \]

### Step 4: Sum of the terms
The sum of the first \( n \) terms of an arithmetic sequence is given by:
\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]
For the first sequence \( s_n \):
\[ S_{10} = \frac{10}{2} \left(2a_1 + 9d_1\right) = 5(2a_1 + 9d_1) \]
For the second sequence \( t_n \):
\[ S_{15} = \frac{15}{2} \left(2a_2 + 14d_2\right) = \frac{15}{2} \left(2a_1 + 14d_2\right) \]
Setting these sums equal gives:
\[ 5(2a_1 + 9d_1) = \frac{15}{2}(2a_1 + 14d_2) \]

### Step 5: Simplify the equation
Multiplying both sides by 2 to eliminate the fraction:
\[ 10(2a_1 + 9d_1) = 15(2a_1 + 14d_2) \]
Expanding both sides:
\[ 20a_1 + 90d_1 = 30a_1 + 210d_2 \]
Rearranging gives:
\[ 20a_1 - 30a_1 + 90d_1 - 210d_2 = 0 \]
This simplifies to:
\[ -10a_1 + 90d_1 - 210d_2 = 0 \]
Dividing through by 10:
\[ -a_1 + 9d_1 - 21d_2 = 0 \]
Thus:
\[ 9d_1 = a_1 + 21d_2 \]

### Step 6: Find the ratio \( \frac{s_2 - s_1}{t_2 - t_1} \)
We need to find:
\[ \frac{s_2 - s_1}{t_2 - t_1} = \frac{(a_1 + d_1) - a_1}{(a_2 + d_2) - a_2} = \frac{d_1}{d_2} \]
From the equations we derived, we have:
\[ d_1 = a_1 + 2d_2 \]
Substituting this into the ratio gives:
\[ \frac{d_1}{d_2} = \frac{a_1 + 2d_2}{d_2} = \frac{a_1}{d_2} + 2 \]

### Step 7: Solve for \( \frac{s_2 - s_1}{t_2 - t_1} \)
Using the earlier derived relationship \( 9d_1 = a_1 + 21d_2 \):
\[ d_1 = \frac{a_1 + 21d_2}{9} \]
Substituting back gives:
\[ \frac{s_2 - s_1}{t_2 - t_1} = \frac{\frac{a_1 + 21d_2}{9}}{d_2} = \frac{a_1 + 21d_2}{9d_2} \]
Substituting \( a_1 = 19d_2 \) gives:
\[ \frac{19 + 21}{9} = \frac{40}{9} \]

### Final Answer
The value of \( \frac{s_2 - s_1}{t_2 - t_1} \) is \( \frac{19}{8} \).