Introduction to Shifted Geometric Sequences (A first look at limits)

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Presentation transcript:

Introduction to Shifted Geometric Sequences (A first look at limits) Learning Targets: I can distinguish between arithmetic, geometric, and shifted geometric sequences. I can use a recursive formula to generate a sequence. I can use technology to simulate arithmetic and geometric sequences. I can determine long run values of geometric and shifted geometric sequences.

Discuss Limits… Interesting Fact: the women’s world record for the fastest time in the 100m dash has decreased by about 3 seconds in 66 years. (currently Florence Griffith-Joyner USA owns the record at 10.49 seconds). An expert predicted that the ultimate performance for a woman in the 100m dash will be 10.15 seconds and not decrease after that.

Limit: A “long-run value” that a sequence or a function approaches Limit: A “long-run value” that a sequence or a function approaches. The quantity that is associated with the point of stability in dynamic systems. Think of it as a “cut off” value. Like the speed limit. Shifted Geometric Sequence – a geometric sequence that includes an added term in the recursive rule.

Example: Antonio and Deanna are working at the community pool for the summer. They need to provide a “Shock” treatment of 450g of dry chlorine to prevent the growth of algae in the pool, then they add 45g of chlorine each day after the initial treatment. Each day the sun burns off 15% of the chlorine. Find the amount of chlorine after 1 day, 2 days, and 3 days. Determine the long run value.

How to determine the Long-run value… Look at the table in your calculator. Scroll down until the values level off. Look at the graph to see where the graph levels off. In your “home” screen: 450 (enter), x(0.85)+45 (enter, enter…) until the values level off. In your “home” screen: find u(50), u(100), u(150), etc.

Example: Find the value of u1, u2, and u3, identify the type of sequence (arithmetic, geometric, or shifted geometric), tell whether it is increasing or decreasing. Lastly, find the long run value for the sequence. u0 = 24 un = (1-0.60) un-1 +30

Example: Find the value of u1, u2, and u3, identify the type of sequence (arithmetic, geometric, or shifted geometric), tell whether it is increasing or decreasing. Lastly, find the long run value for the sequence. u0 = 434 un = (1-0.09) un-1

Assignment: pg. 48 1, 3, 5, 9 Worksheet: 1 & 2