Number sequences

Learning Objectives Identify when the parameters of an ellipse form an arithmetic or geometric sequence. Determine the equation of the nth ellipse in a defined sequence. Calculate the area of a specific ellipse within a sequence. Find the sum of the areas of the first n ellipses in a sequence by applying series formulas. Analyze the change in eccentricity for a sequence of ellipses. Model scenarios involving sequences of expanding or shrinking ellipses. Distinguish between sequences of ellipses defined by arithmetic vs. geometric progressions. Imagine a ripple in a pond, but instead of perfect circles, the waves are ellipses that grow in a predictable pattern. How could we mathematically describe the 10th ripple? 🌊 This tutorial merges two key Grade 11 topics: the predic...

TermDefinitionExample Elliptical SequenceA sequence where each term is an ellipse, and one or more of its parameters (e.g., semi-major axis, semi-minor axis, focal distance) change according to a number sequence rule.A sequence of ellipses where the semi-major axis lengths are 5, 10, 15, 20... and the semi-minor axis lengths are 2, 4, 6, 8... Arithmetic Elliptical SequenceAn elliptical sequence where a key parameter, like the semi-major axis (a) or semi-minor axis (b), increases or decreases by a constant amount (a common difference, d) for each subsequent term.The semi-major axes of a sequence of ellipses are 10, 13, 16, 19... Here, a₁ = 10 and the common difference d = 3. Geometric Elliptical SequenceAn elliptical sequence where a key parameter is multiplied by a constant factor (a comm...

Equation of the nth Ellipse \frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1 To find the equation of the nth ellipse in a sequence, first find the values of its semi-major axis (a_n) and semi-minor axis (b_n) using the appropriate sequence formulas: a_n = a_1 + (n-1)d (arithmetic) or a_n = a_1 \cdot r^{n-1} (geometric). Then substitute these values into the standard ellipse equation. Area of the nth Ellipse A_n = \pi a_n b_n Once you have determined the semi-axes a_n and b_n for the nth ellipse, use this formula to calculate its specific area. This is the first step before finding the sum of a series of areas. Sum of Areas (Geometric Case) S_n = A_1 \frac{1 - (r_{area})^n}{1 - r_{area}} If the semi-axes a_n and b_n both follow a geometric progression with a common ratio &#0...