Common Difference

Algebra 2 · Axiom Academy

LESSON Arithmetic Sequences: The Common Difference One fixed step, called the common difference, decides everything an arithmetic sequence does. 1. A Constant Difference, Not a Constant Ratio What makes a sequence arithmetic ? The jump from each term to the next is an addition of the same number , so the difference between any term and the one before it never changes. Contrast that with a geometric sequence, where it is the ratio that stays constant. Arithmetic — the difference is fixed Geometric — the ratio is fixed 2. Finding d : Subtract Each Term From the Next To pin down the common difference, take any term and subtract the term immediately before it. Because the difference is constant, it does not matter which pair you choose — every consecutive pair gives back the same d . Slide from pair to pair and the gap never budges: 3. The Explicit Formula: a_n = a_1 + (n-1)d To reach the n -th term, start at the first term a_1 and add d over and over. The catch is the count : getting from a_1 to a_n takes exactly n-1 additions — so the number multiplying d is always one less than the term number, never n itself. Building 3, 7, 11, 15, 19 from a_1 = 3 , d = 4 Each new term adds one more copy of 4 , and the coefficient counts those copies: term 5 needs 5-1 = 4 of them. Here a_1 is the first term, d is the common difference, and n is the term number you want. 4. What the Common Difference Controls

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