To find the first term $a_1$ in the arithmetic sequence given that the common difference $d = \frac{1}{2}$ and the 15th term $a_{15} = 7$, we can use the formula for the $n$th term of an arithmetic sequence:

$a_n = a_1 + (n-1) \cdot d$

Given:

• $d = \frac{1}{2}$
• $a_{15} = 7$
• $n = 15$

Step 1: Substitute the values into the formula for the 15th term:

$a_{15} = a_1 + (15-1) \cdot \frac{1}{2}$

$7 = a_1 + 14 \cdot \frac{1}{2}$

Step 2: Simplify the equation:

$7 = a_1 + \frac{14}{2}$ $7 = a_1 + 7$

Step 3: Solve for $a_1$:

$a_1 = 7 - 7 = 0$

Answer:

The first term $a_1$ of the arithmetic sequence is 0.

Would you like any further explanations or have any other questions? Here are some related practice problems:

1. What would be the 10th term of this sequence?
2. How would you find the sum of the first 15 terms of this sequence?
3. What is the general formula for the $n$th term in this sequence?
4. How does changing the common difference to $\frac{3}{2}$ affect the first term if the 15th term remains 7?
5. If $a_{15} = 7$ and $d = 1/3$, what would the first term be?

Tip: Always double-check your substitution and arithmetic when solving for the first term in sequences, as small errors can lead to incorrect results.