Identify the behaviour of a sequence

The table begins at index 0. What is the starting term value?

The sequence starts at 3. Try changes of −3, 0 and +3 per step. Observe the discrete points and compare the initial term with later terms

Which feature stays fixed when only the step changes?

A sequence has ordered positions

A sequence assigns one value to each index. In this course the indices are 0, 1, 2, 3, … A subscript is an index, not a power

1. 1Discrete graph Plot one point for each integer index. Do not join the points as though every intermediate index were part of this sequence
2. 2Do not confuse the two numbers For t₀ = 3, the index is 0 and the term value is 3. A negative term value is allowed even though the indices are non-negative

Starting value and repeated change

\(\displaystyle t_0=3,\quad d=-2\)

Starting at 3, subtract 2 each time. What is the term at index 4?

Every term equals 5. How does the sequence behave?

Assume the add-3 pattern continues: −8, −5, −2, 1, … . Which description is correct?

Check the whole displayed pattern

One pair does not establish the direction of all later terms. Check every pair shown and the rule that is assumed to continue

1. 1Increasing or decreasing These labels describe repeated rises or repeated falls, not whether the values are positive or negative
2. 2Oscillating The direction changes repeatedly. The values may repeat, grow in magnitude or shrink in magnitude

Assume the two values keep repeating: 4, 9, 4, 9, … . Which label fits?

Direction and limit are different questions

A sequence can decrease without a finite limit, or oscillate while approaching a limit. Constant sequences also have limits

1. 1Approach from either side Later terms must settle near one fixed value. They do not have to stay on one side of it
2. 2State the assumption For the following models, the stated multiplier is used indefinitely. This is not a claim that a real-world process lasts forever

Start at 8 and repeatedly multiply by r. Compare r = 1, −1, 0.5 and −0.5. Also inspect r = 0, when every term after the first is zero

At r = −0.5, what happens if the rule continues?

With t₀ = 8 and multiplier 0.5 used forever, which statement is correct?

With t₀ = 8 and multiplier −1 used forever, is there a limiting value?

Every term equals 5. What is its limiting value?

Start at 5 and subtract 3 indefinitely. Does this mathematical sequence have a finite limit?

Investigate and explain

Choose a starting value and a repeated rule. Make a table, plot separate points and explain the behaviour. State what you assume happens after the displayed terms

1. 1Use evidence Give consecutive values that support the direction. For a limit, explain what the rule does to later terms
2. 2Keep the model honest Counts cannot be fractional or negative. A mathematical continuation may outlast the useful real-world model. Your graph and explanation need self or teacher review