lesson
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If a sequence starts at 7 and jumps by 4 every single step, what is the 500th number? You definitely don't want to hit "+4" on your calculator 499 times.
The Logic of the Jumps
An arithmetic sequence is simply a list of numbers where each term is found by adding a constant value called the common difference (d) to the previous term.
Let's watch how the terms build from the very first number, which we call a1.
📊Interactive diagram
Notice the counting trick: to reach the 2nd term, you take 1 jump. To reach the 4th term, you take 3 jumps.
So to get from term 1 to any term n, how many jumps of size d must you take?
Deriving the Formula
Because you start at the first term, you always make exactly n−1 jumps to reach position n.
This gives us the explicit nth-term formula: an=a1+(n−1)d
📊Interactive diagram
Mathematicians across ancient civilizations, including the 7th-century Indian scholar Brahmagupta, formalized these linear progression rules to calculate astronomical cycles and commerce debts without tedious manual addition.
How does this look with real numbers when the sequence goes down instead of up?