An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. Series Purplemath The two simplest sequences to work with are arithmetic and geometric sequences. You may wish to use graphing software such as the free-to-download Geogebra to investigate the graphs. Arithmetic & Geometric Sequences Intro Examples Arith. In an arithmetic series the terms change by a. What if the starting number for your geometric sequence is a fraction, or a negative number? In addition to binomial expansion with negative/fractional powers, you will study arithmetic and geometric series. What if the common ratio is a fraction, or a negative number? Here are some questions you might like to explore:Ĭan you make any predictions about the graph from the geometric sequence you use to generate the equation? If the sum of the roots of the quadratic equation ax2 +bx+ c 0 is equal to the sum. Each term after the first term is obtained by multiplying the previous term by r, the common ratio. Oliver's sequence starts at $1$ and has common ratio $2$ (each number in the sequence is $2$ times the previous number).Ĭreate some more geometrical sequences and substitute consecutive terms into Oliver's quadratic equation. 8.2: Problem Solving with Arithmetic Sequences 8.4: Quadratic Sequences Jennifer Freidenreich Diablo Valley College Geometric sequences have a common ratio. Oliver's sequence is an example of a geometrical sequence, created by taking a number and then repeatedly multiplying by a common ratio. Oliver has been experimenting with quadratic equations of the form: $$y=ax^2+2bx+c$$ Oliver chose values of $a, b$ and $c$ by taking three consecutive terms from the sequence: $$1, 2, 4, 8, 16, 32.$$ Try plotting some graphs based on Oliver's quadratic equations, for different sets of consecutive terms from his sequence.Ĭan you make any generalisations? Can you prove them?
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