The sum of the 3rd and 7th terms of an AP is 6, and their product is 8. Given that the AP is increasing, find its first term.

Two numbers with sum 6 and product 8 are 2 and 4. Since the AP increases, $a_3=2$ and $a_7=4$. Then $a_7-a_3=4d=2$, so $d=0.5$, and $a=a_3-2d=2-1=1$. The value 2 is the 3rd term mistaken for the first, 4 is the 7th term, and 0.5 is the common difference.