\]Question: A primatologist observes that the number of vocalizations made by a troop of chimpanzees follows an arithmetic sequence over five consecutive days. If the total number of vocalizations over the five days is 120, and the difference between the number of vocalizations on the last day and the first day is 16, what is the number of vocalizations on the third day?

["Question: A primatologist observes that the number of vocalizations made by a troop of chimpanzees follows an arithmetic sequence over five consecutive days. If the total number of vocalizations over the five days is 120, and the difference between the number of vocalizations on the last day and the first day is 16, what is the number of vocalizations on the third day?**", "### Understanding the Arithmetic Sequence", "Let the number of vocalizations on the five consecutive days be represented by the arithmetic sequence:", "$$\na, a + d, a + 2d, a + 3d, a + 4d\n$$", "Where:\n- $ a $ is the number of vocalizations on the first day,\n- $ d $ is the common difference.", "### Total Vocalizations Over Five Days", "The sum of an arithmetic sequence is given by:", "$$\nS_n = \frac{n}{2}(2a + (n-1)d)\n$$", "For $ n = 5 $, the total is 120:", "$$\n\frac{5}{2}(2a + 4d) = 120\n$$", "Multiply both sides by 2:", "$$\n5(2a + 4d) = 240\n$$", "$$\n2a + 4d = 48 \quad \ ext{(Equation 1)}\n$$", "### Difference Between Last and First Day", "We are told the difference between the last day and the first day is 16:", "$$\n(a + 4d) - a = 16 \Rightarrow 4d = 16 \Rightarrow d = 4\n$$", "### Substitute $ d = 4 $ into Equation 1", "$$\n2a + 4(4) = 48 \Rightarrow 2a + 16 = 48 \Rightarrow 2a = 32 \Rightarrow a = 16\n$$", "### Find the Number of Vocalizations on the Third Day", "The third day corresponds to the term $ a + 2d $:", "$$\na + 2d = 16 + 2(4) = 16 + 8 = 24\n$$", "Thus, the number of vocalizations on the third day is $\boxed{24}$"]