Question

Find the twelfth term of an AP with first term -1 and common difference 4

Short answer

The twelfth term of the AP is 43.

Step-by-step solution

Identify the First Term and the Common Difference

The first term (denoted as a1) of the arithmetic progression is given as -1, and the common difference (d) is given as 4. This means each subsequent term in the sequence increases by 4.

Use the Arithmetic Progression Formula

To find the nth term of an arithmetic progression, the formula is a_n = a_1 + (n-1)*d where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.

Calculate the Twelfth Term

We need to find the twelfth term (a_12). Replace n with 12 in the formula: a_12 = a_1 + (12-1)*d = -1 + 11*4.

Solve for the Twelfth Term

Calculate the value of a_12: a_12 = -1 + 44 = 43. Thus, the twelfth term of the arithmetic progression is 43.

Key concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a series of numbers in which each term after the first is obtained by adding a constant number, called the common difference, to the previous term. This sequence is characterized by the regularity in which every term increments or decrements.

For example, if you start with the number 5 and keep adding 3, your arithmetic sequence would start as 5, 8, 11, 14, and so on. It is easy to predict any number in the sequence once you know the first term and the common difference. In the context of homework or textbook exercises, mastering arithmetic sequences allows students to tackle problems involving regular patterns or intervals efficiently.

Common Difference

The common difference is the backbone of an arithmetic sequence. It is the constant difference between consecutive terms in the sequence. To identify this value, simply subtract one term from the next term. It's important to note that if the common difference is positive, the sequence will increase, and if it's negative, the sequence will decrease.

This feature facilitates the process of forecasting future terms or finding past terms within the sequence. Given the textbook example, the first term is -1, and the sequence increments by 4, thus establishing 4 as the common difference. Recognizing the common difference is crucial for solving many problems related to arithmetic sequences.

Sequence and Series

In mathematics, a sequence is an ordered list of numbers and a series is the sum of the terms of a sequence. An arithmetic sequence refers to the ordered list where each term differs by a constant value (the common difference), while an arithmetic series would be the sum of the terms of such a sequence.

Understanding sequences and series is vital as it extends to various applications, from simple interest in banking to concepts in physics like motion. It is also foundational knowledge for more advanced mathematical topics. Taking time to understand the nature of sequences not only helps with homework exercises but builds a solid base for future learning in mathematics and its related fields.