AI Tutor
Imagine finding a torn receipt where only two line items survived: the 4th4^{\text{th}} item cost \19andtheand the9^{\text{th}}itemcostitem cost$44$.
If every item increases by the exact same price, how do you determine the price of the very first item or find out which item hit \104$?

The Distance Between Terms

An arithmetic sequence is a list of numbers where the step between any two consecutive values is a constant constant called the common difference (dd).
Moving from the 1st1^{\text{st}} term a1a_1 to the nthn^{\text{th}} term ana_n takes exactly nβˆ’1n - 1 steps of size dd, giving the explicit formula an=a1+(nβˆ’1)da_n = a_1 + (n - 1)d.
What if you do not know the first term, but only two arbitrary terms somewhere in the middle?

Finding the Common Difference

To step from any term aka_k to a later term ama_m, you must take exactly mβˆ’km - k jumps of size dd.
This gives the relation am=ak+(mβˆ’k)da_m = a_k + (m - k)d, which we can rearrange to find dd directly: d=amβˆ’akmβˆ’kd = \frac{a_m - a_k}{m - k}
This formula mirrors the slope of a linear function Ξ”yΞ”x\frac{\Delta y}{\Delta x}, because an arithmetic sequence is simply a linear function evaluated at integer inputs.
Let's apply this to our opening challenge: a4=19a_4 = 19 and a9=44a_9 = 44.