What is the difference between a sequence and a series in IB DP Mathematics?

In IB DP Mathematics, a sequence is an ordered list of numbers following a specific pattern, while a series is the sum of the terms of a sequence. For example, in the sequence 2, 4, 6, 8, the series would be the sum 2 + 4 + 6 + 8.

How do you find the nth term of an arithmetic sequence?

To find the nth term of an arithmetic sequence, use the formula: a_n = a_1 + (n-1)d, where a_n is the nth term, a_1 is the first term, and d is the common difference between consecutive terms. This formula helps in identifying any term in the sequence based on its position.

What is the formula for the sum of an arithmetic series?

The sum of the first n terms of an arithmetic series can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where S_n is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the nth term. This formula is crucial for solving problems involving the total of a sequence of numbers.

How do you determine if a sequence is geometric?

A sequence is geometric if the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted as r. For example, in the sequence 3, 6, 12, 24, the common ratio is 2, as each term is obtained by multiplying the previous term by 2.

What is the formula for the sum of a geometric series?

The sum of the first n terms of a geometric series is given by the formula: S_n = a_1 * (1 - r^n) / (1 - r), where S_n is the sum, a_1 is the first term, r is the common ratio, and n is the number of terms. This formula is applicable when the common ratio r is not equal to 1.

Sequence and Series

Summary and Exam Tips for Sequence and Series

Sequence and Series is a subtopic of Sequences and Series, which falls under the subject Mathematics in the IB DP curriculum. A sequence is an infinite list of numbers, often defined as a function from the set of whole numbers $\mathbb{N}$ to the set of real numbers $\mathbb{R}$ . Examples include arithmetic sequences, where each term increases by a constant difference, and geometric sequences, where each term is multiplied by a constant ratio. The Fibonacci sequence is a special type of sequence defined recursively.

In an arithmetic sequence, the general term $T_n$ is given by $T_n = a + (n-1)d$ , where $a$ is the first term and $d$ is the common difference. For geometric sequences, the general term is $T_n = ar^{n-1}$ , with $a$ as the first term and $r$ as the common ratio.

A series is the sum of the terms of a sequence. For arithmetic series, the sum of the first $n$ terms is $S_n = \frac{n}{2}(2a + (n-1)d)$ . For geometric series, the sum is $S_n = \frac{a(1-r^n)}{1-r}$ for $r \neq 1$ . The Binomial Theorem provides a way to expand expressions of the form $(x+y)^n$ using binomial coefficients.

Exam Tips

Understand Definitions: Clearly understand the definitions of sequences and series, including arithmetic and geometric sequences. This foundational knowledge is crucial for solving problems.
Practice Formulas: Familiarize yourself with the formulas for the general terms and sums of arithmetic and geometric sequences. Practice applying these formulas to different problems.
Recursive Sequences: Pay attention to recursive sequences like the Fibonacci sequence. Practice deriving terms and sums using recursive relations.
Binomial Theorem: Practice expanding expressions using the Binomial Theorem. Understand how to calculate binomial coefficients and apply them in expansions.
Problem-Solving: Work through various examples and past exam questions to strengthen your problem-solving skills. This will help you recognize patterns and apply the correct formulas quickly during exams.