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. 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