The 6th term of an arithmetic sequence is \( x \) while the 11 th term is \( y \). find the Ist 2 terms.

To determine the first two terms of the arithmetic sequence given that the 6th term is \( x \) and the 11th term is \( y \), let's follow these steps: ### 1. Understanding the Arithmetic Sequence An arithmetic sequence is defined by its first term \( a \) and a common difference \( d \) between consecutive terms. The \( n \)-th term of an arithmetic sequence can be expressed as: \[ a_n = a + (n - 1)d \] ### 2. Setting Up the Equations Given: - The 6th term (\( a_6 \)) is \( x \): \[ a + 5d = x \quad \text{(1)} \] - The 11th term (\( a_{11} \)) is \( y \): \[ a + 10d = y \quad \text{(2)} \] ### 3. Solving for the Common Difference \( d \) Subtract equation (1) from equation (2): \[ (a + 10d) - (a + 5d) = y - x \\ 5d = y - x \\ d = \frac{y - x}{5} \] ### 4. Solving for the First Term \( a \) Substitute \( d \) back into equation (1): \[ a + 5\left(\frac{y - x}{5}\right) = x \\ a + (y - x) = x \\ a = x - (y - x) \\ a = 2x - y \] ### 5. Determining the Second Term The second term of the sequence is: \[ a_2 = a + d = (2x - y) + \frac{y - x}{5} \] Simplify: \[ a_2 = \frac{10x - 5y + y - x}{5} = \frac{9x - 4y}{5} \] ### **Final Answer** - **First Term (\( a \))**: \( 2x - y \) - **Second Term (\( a_2 \))**: \( \frac{9x - 4y}{5} \) --- **Example:** If \( x = 15 \) and \( y = 30 \): - First Term: \( 2(15) - 30 = 0 \) - Second Term: \( \frac{9(15) - 4(30)}{5} = \frac{135 - 120}{5} = 3 \) Thus, the first two terms would be \( 0 \) and \( 3 \).