1. **State the problem:** We need to find the first term $a$ and the common difference $p$ of an arithmetic progression (AP) given that the 5th term is 20 and the 10th term is 40. 2. **Recall the formula for the $n$th term of an AP:** $$a_n = a + (n-1)p$$ where $a$ is the first term and $p$ is the common difference. 3. **Write equations for the given terms:** - For the 5th term: $$a_5 = a + 4p = 20$$ - For the 10th term: $$a_{10} = a + 9p = 40$$ 4. **Subtract the first equation from the second to eliminate $a$:** $$a + 9p - (a + 4p) = 40 - 20$$ $$\cancel{a} + 9p - \cancel{a} - 4p = 20$$ $$5p = 20$$ 5. **Solve for $p$:** $$p = \frac{20}{5} = 4$$ 6. **Substitute $p=4$ back into the first equation to find $a$:** $$a + 4 \times 4 = 20$$ $$a + 16 = 20$$ $$a = 20 - 16 = 4$$ **Final answer:** $$a = 4, \quad p = 4$$