AI Tutor
If you save 50inyourfirstmonthandaddanextra50 in your first month and add an extra 15 every month after that, how much will you deposit in month 100?
Instead of adding 15ninety−ninetimesinarow,wecanmodelthiswithan∗∗arithmeticprogression∗∗—asequencewherethedifferencebetweenanytwoconsecutivetermsisaconstantvaluecalledthe∗∗commondifference∗∗(15 ninety-nine times in a row, we can model this with an **arithmetic progression**—a sequence where the difference between any two consecutive terms is a constant value called the **common difference** (d$).
How do we jump straight to any distant term without listing the ones in between?

The nth Term Formula

Around 300 BCE, Babylonian astronomers tracked regular planetary steps across the night sky using constant step sequences to predict celestial events.
Notice the structure: the first term is u1=au_1 = a. To get to u2u_2, you add one difference (a+da + d); to reach u3u_3, you add two differences (a+2da + 2d).
Because reaching the nthn^{\text{th}} term requires starting at aa and taking exactly n−1n - 1 steps of size dd, our general formula is un=a+(n−1)du_n = a + (n - 1)d
Now let's put this formula to work on a concrete numerical sequence.

Finding Terms and Testing Membership

Consider the sequence 7,11,15,19,…7, 11, 15, 19, \dots. Here, the first term is a=7a = 7 and the common difference is d=11−7=4d = 11 - 7 = 4.
To find the 50th term, substitute n=50n = 50: u50=7+(50−1)(4)=7+49(4)=203u_{50} = 7 + (50 - 1)(4) = 7 + 49(4) = 203