Geometric and Arithmetic Sequences

Running Head: Geometric and Arithmetic Sequences

Geometric and Arithmetic Sequences

Cindy Giroux

MAT126: Survey of Mathematical Methods

Jake Grandy

September 26, 2012

Running Head: Geometric and Arithmetic Sequences

Since arithmetic and geometric sequences are so nice and regular, they have formulas. For arithmetic sequences, the common difference is d, and the first term a1 is often referred to simply as "a". Since you get the next term by adding the common difference, the value of a2 is just a + d. The third term is a3 = (a + d) + d = a + 2d. The fourth term is a4 = (a + 2d) + d = a + 3d. Following this pattern, the n-th term an will have the form an = a + (n – 1)d.

An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. For instance, 2, 5, 8, 11, 14,... and 7, 3, –1, –5,... are arithmetic, since you add 3 and subtract 4, respectively, at each step. A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16,... and 81, 27, 9, 3, 1, 1/3,... are geometric, since you multiply by 2 and divide by 3, respectively, at each step.

The number added (or subtracted) at each stage of an arithmetic sequence is called the "common difference" d, because if you subtract (find the difference of) successive terms, you'll always get this common value. The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (find the ratio of) successive terms, you'll always get this common value.

A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16,... and 81, 27, 9, 3, 1, 1/3,... are geometric, since you multiply by 2 and divide by 3, respectively, at each step.

For geometric sequences, the common ratio is r, and the first term a1 is often referred to simply as "a". Since you get the next...