Number sequences involving decimals

Learning Objectives Identify arithmetic and geometric sequences involving decimal terms. Determine the common difference or common ratio in a decimal sequence. Formulate an explicit formula (nth term rule) for a given decimal sequence. Apply the nth term rule to find a specific term in a sequence. Solve for the position 'n' of a given decimal term in a sequence. Analyze and extend sequences with alternating or multi-step decimal patterns. Justify the classification of a sequence as arithmetic, geometric, or neither based on logical deduction. Ever noticed how a bouncing ball's height decreases by a certain fraction each time? 🏀 That's a real-world decimal sequence in action! In this tutorial, we will explore the logic behind number sequences that use...

TermDefinitionExample SequenceAn ordered list of numbers, called terms, that follow a specific logical rule or pattern.1.2, 2.4, 3.6, 4.8, ... Term (a_n)An individual number in a sequence. The subscript 'n' indicates its position in the sequence.In the sequence 0.5, 1.0, 1.5, ..., the third term is denoted as a_3, and its value is 1.5. Arithmetic SequenceA sequence where the difference between any two consecutive terms is a constant value.The sequence 3.1, 3.4, 3.7, 4.0, ... is arithmetic because 0.3 is added to each term to get the next. Common Difference (d)The constant value that is added to each term to get the next term in an arithmetic sequence. It can be positive, negative, or zero.In the sequence 10.5, 10.0, 9.5, ..., the common difference (d) is -0.5. Geometric Sequence...

Arithmetic Sequence nth Term Formula a_n = a_1 + (n-1)d Use this to find any term (a_n) in an arithmetic sequence, given the first term (a_1), the term's position (n), and the common difference (d). Geometric Sequence nth Term Formula a_n = a_1 * r^(n-1) Use this to find any term (a_n) in a geometric sequence, given the first term (a_1), the term's position (n), and the common ratio (r). Logical Test for an Arithmetic Sequence d = a_n - a_(n-1) To prove a sequence is arithmetic, demonstrate that the difference between any term and its preceding term is a constant value, d. Logical Test for a Geometric Sequence r = a_n / a_(n-1) To prove a sequence is geometric, demonstrate that the ratio of any term to its preceding term is a constant value, r.