GR10 Mathematics - Sequences and Series (Vocabulary)

Vocabulary flashcards covering key sequence and series terms from the notes.

Sequence

A function whose domain is a finite set {1,2,…,n} or an infinite set {1,2,…}; terms are usually denoted a1, a2, …, a_n.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant (the common difference d).

Common difference (d)

The constant amount added to each term to obtain the next term in an arithmetic sequence.

First term (a1)

The initial term of a sequence.

Nth term (a_n)

The general term of a sequence; for arithmetic sequences a_n = a1 + (n - 1)d.

Arithmetic mean

The terms inserted between two nonconsecutive terms of an arithmetic sequence to create evenly spaced terms.

Arithmetic series

The sum of the terms of an arithmetic sequence; formulas: Sn = n/2 [2a1 + (n-1)d] or Sn = n/2 (a1 + a_n).

Geometric sequence

A sequence where each term after the first is obtained by multiplying by a nonzero constant called the common ratio.

Common ratio (r)

The fixed constant multiplied by each term to obtain the next term; r = a{n+1} / an.

Nth term of a geometric sequence

a_n = a1 · r^(n-1), where a1 is the first term and r is the common ratio.

Geometric mean

The nth root of the product of a set of numbers; a measure of central tendency for multiplicative data.

Geometric series

The sum of the terms of a geometric sequence.

Sum of geometric series (r ≠ 1)

S_n = a1 (1 − r^n) / (1 − r).

Remainder Theorem

If P(x) is divided by (x − r), the remainder equals P(r).

Factor Theorem

P(x) has a factor (x − r) if and only if P(r) = 0.

Sequence

A function whose domain is a finite set {1,2,…,n} or an infinite set {1,2,…}; terms are usually denoted a1, a2, \dots, a_n.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant (the common difference d).

Common difference (d)

The constant amount added to each term to obtain the next term in an arithmetic sequence; it can be found using the formula d = a{n+1} - an.

First term (a_1)

The initial term of a sequence.

Nth term (a_n)

The general term of a sequence; for arithmetic sequences, the formula is an = a1 + (n - 1)d.

Arithmetic mean

The terms inserted between two nonconsecutive terms of an arithmetic sequence to create evenly spaced terms. For two numbers a and b, the arithmetic mean is (a+b)/2.

Arithmetic series

The sum of the terms of an arithmetic sequence; formulas: Sn = \frac{n}{2} [2a1 + (n-1)d] or Sn = \frac{n}{2} (a1 + a_n).

Geometric sequence

A sequence where each term after the first is obtained by multiplying by a nonzero constant called the common ratio.

Common ratio (r)

The fixed constant multiplied by each term to obtain the next term; r = a{n+1} / an.

Nth term of a geometric sequence

an = a1 \cdot r^{n-1}, where a_1 is the first term and r is the common ratio.

Geometric mean

The nth root of the product of a set of numbers; a measure of central tendency for multiplicative data. For two numbers a and b, the geometric mean is \sqrt{ab} .

Geometric series

The sum of the terms of a geometric sequence.

Sum of geometric series (r
1)

Sn = a1 (1 - r^n) / (1 - r).

Remainder Theorem

If P(x) is divided by (x - r), the remainder equals P(r).

P(x) has a factor (x - r) if and only if P(r) = 0.