Question:

**Step 1:** Let the arithmetic sequence have first term $a$ and common difference $d$. The sum of the first 10 terms, $S_{10}$, is given by the formula $S_n = \frac{n}{2}(2a + (n-1)d)$. $$S_{10} = \frac{10}{2}(2a + (10-1)d) = 5(2a + 9d) = 10a + 45d$$ The 58th term, $a_{58}$, is given by the formula $a_n = a + (n-1)d$. $$a_{58} = a + (58-1)d = a + 57d$$ **Step 2:** Set up the given condition that the sum of the first ten terms is equal to the 58th term. $$S_{10} = a_{58}$$ $$10a + 45d = a + 57d$$ Subtract $a$ and $45d$ from both sides: $$9a = 12d$$ Divide by 3: $$3a = 4d$$ **Step 3:** Determine the properties of $a$ and $d$. The problem states that the sequence consists of positive integers. This implies that the first term $a$ must be a positive integer ($a \in \mathbb{Z}^+$). For all terms $a_n = a + (n-1)d$ to be integers, the common difference $d$ must also be an integer. From the equation $3a = 4d$, since $a$ is a positive integer, $3a$ is a positive integer. Therefore, $4d$ must be a positive integer, which implies $d$ must be a positive integer ($d \in \mathbb{Z}^+$). Since $3a = 4d$ and 3 and 4 are coprime, $a$ must be a multiple of 4, and $d$ must be a multiple of 3. Let $a = 4k$ for some positive integer $k$. Substitute this into the equation $3a = 4d$: $$3(4k) = 4d$$ $$12k = 4d$$ $$d = 3k$$ Since $a$ and $d$ must be positive integers, $k$ must be a positive integer. The smallest possible positive integer value for $k$ is $k=1$. **Step 4:** Find the least possible value of the first term. To find the least possible value of $a$, we use the smallest possible positive integer value for $k$, which is $k=1$. Substituting $k=1$ into the expressions for $a$ and $d$: $$a = 4(1) = 4$$ $$d = 3(1) = 3$$ We verify that this choice of $a$ and $d$ results in a sequence of positive integers: The first term is $a=4$, which is a positive integer. The common difference is $d=3$, which is a positive integer. The general term is $a_n = a + (n-1)d = 4 + (n-1)3 = 4 + 3n - 3 = 3n+1$. For any $n \geq 1$, $3n+1$ is a positive integer. Thus, the sequence $4, 7, 10, \dots$ consists entirely of positive integers. Therefore, the least possible value of the first term is 4.