We are given an arithmetic sequence $\{a_n\}$ with $a_{10} = 10$ and $S_{10} = 70$. We need to find the common difference $d$.

1. Problem Description

We are given an arithmetic sequence $\{a_n\}$ with $a_{10} = 10$ and $S_{10} = 70$. We need to find the common difference $d$.

2. Solution Steps

The sum of the first $n$ terms of an arithmetic sequence is given by:

$S_n = \frac{n(a_1 + a_n)}{2}$.

In our case, $n = 10$ and $S_{10} = 70$, so we have

$S_{10} = \frac{10(a_1 + a_{10})}{2} = 70$.

We are given $a_{10} = 10$, so we can substitute it into the equation:

$\frac{10(a_1 + 10)}{2} = 70$

$5(a_1 + 10) = 70$

$a_1 + 10 = 14$

$a_1 = 4$.

The $n$-th term of an arithmetic sequence is given by:

$a_n = a_1 + (n-1)d$.

In our case, $a_{10} = 10$, so we have

$a_{10} = a_1 + (10-1)d = a_1 + 9d$.

We know $a_1 = 4$ and $a_{10} = 10$, so we can substitute them into the equation:

$10 = 4 + 9d$

$6 = 9d$

$d = \frac{6}{9} = \frac{2}{3}$.

3. Final Answer

The common difference $d = \frac{2}{3}$.