Geometric sequences with fractions

Learning Objectives Identify a geometric sequence where the terms are angle measures. Determine the fractional common ratio of a geometric sequence of angles. Calculate the nth term of a geometric sequence of angles using the formula a_n = a_1 * r^(n-1). Model real-world scenarios, like repeated angle bisection, using geometric sequences. Solve problems connecting geometric sequences to iterative geometric constructions, such as fractals. Distinguish between the number of iterations and the term number 'n' in a sequence. Accurately compute powers of fractions as required by the geometric sequence formula. Ever seen a fractal pattern and wondered how its intricate details are formed? ❄️ It's often built by repeating a simple action, like shrinking an angle by...

TermDefinitionExample Geometric SequenceA sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.A sequence of angle measures starting at 80° and repeatedly being halved: 80°, 40°, 20°, 10°, ... Common Ratio (r)The constant factor between consecutive terms in a geometric sequence. In our context, it will be a fraction, indicating that the angles are decreasing in size.In the sequence 80°, 40°, 20°, ..., the common ratio is r = 40/80 = 1/2. Angle BisectionThe process of dividing an angle into two equal, adjacent angles.Bisecting a 90° right angle creates two 45° angles. This action corresponds to a common ratio of r = 1/2. Term (a_n)A specific element in a sequence. a_1 is the first term, a_2 is the s...

Nth Term of a Geometric Sequence a_n = a_1 * r^(n-1) Use this formula to find the value of any specific term (a_n) in a geometric sequence. You need to know the first term (a_1), the common ratio (r), and the term number (n) you want to find. Finding the Common Ratio r = a_n / a_(n-1) To find the common ratio, divide any term by its preceding term. For example, r = a_2 / a_1 or r = a_3 / a_2. This is useful for confirming a sequence is geometric and finding the value of 'r'.