The 6th term of an arithmetic sequence is while the 11 th term is .
find the Ist 2 terms.

Question

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

UpStudy AI · Accepted Answer

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 $$ 3 $$.
The first term is $$ 2x - y $$ and the second term is $$ \frac{9x - 4y}{5} $$.

The 6th term of an arithmetic sequence is while the 11 th term is .
find the Ist 2 terms.

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The 6th term of an arithmetic sequence is while the 11 th term is . find the Ist 2 terms.

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The 6th term of an arithmetic sequence is while the 11 th term is .
find the Ist 2 terms.