Geometric Series - Key Terms and Formulas

Vocabulary flashcards covering the concepts needed to find the sum of a geometric sequence as shown in the notes.

Geometric sequence

A sequence in which each term after the first is obtained by multiplying the previous term by a constant nonzero value called the common ratio (r).

First term (a1)

The initial term of a geometric sequence.

Common ratio (r)

The constant factor by which consecutive terms are multiplied to obtain the next term.

Nth term (a_n)

The general term of a geometric sequence: a_n = a1 · r^(n-1).

Number of terms (n)

The total number of terms to be considered (e.g., in a partial sum).

Sum of first n terms (S_n)

The total of the first n terms of a geometric sequence.

Sum formula (r ≠ 1)

S_n = a1(1 − r^n)/(1 − r) = a1(r^n − 1)/(r − 1).

Geometric sequence

A sequence in which each term after the first is obtained by multiplying the previous term by a constant nonzero value called the common ratio (r).

First term (a1)

The initial term of a geometric sequence.

Common ratio (r)

The constant factor by which consecutive terms are multiplied to obtain the next term.

Nth term (a_n)

The general term of a geometric sequence: an = a1 \cdot r^{(n-1)}.

Number of terms (n)

The total number of terms to be considered (e.g., in a partial sum).

Sum of first n terms (S_n)

The total of the first n terms of a geometric sequence.

Sum formula (r \neq 1)

Sn = \frac{a1(1 - r^n)}{(1 - r)} = \frac{a_1(r^n - 1)}{(r - 1)}.

Solving for the n^{th} term (a_n)

To find a specific term in a geometric sequence, use the formula an = a1 \cdot r^{(n-1)} given the first term (a_1), common ratio (r), and the term's position (n).

Solving for the sum of the first n terms (S_n)

To calculate the total sum of a given number of terms, use the formula Sn = \frac{a1(1 - r^n)}{(1 - r)} (valid for r \neq 1), requiring the first term (a_1), common ratio (r), and the number of terms (n).

Solving for the common ratio (r)

1. If consecutive terms are known (ak and a{k-1}), divide a term by its preceding term (r = ak / a{k-1}). 2. If the first term (a1) and another term (an) are known, use the an formula (an = a1 \cdot r^{(n-1)}) and solve for r using r = (an / a_1)^{1/(n-1)}.

If the common ratio (r), a term ($$a_