A sequence is a function whose inputs are indices, usually integers. A recursive rule tells you how to get the next value from an earlier one; an explicit rule tells you the value directly from its index. Both need a starting index and enough initial information. Without that information, a recurrence such as “add five” describes many possible sequences.
Predict
Predict how many changes separate index 1 from index n. There are n−1, not n. If the sequence instead starts at zero, there are n changes. Predict whether joining plotted sequence points defines the original sequence at fractional indices. It creates an extension; those new inputs were not part of the original domain.
Experiment
Graph D: discrete sequence. Only the points (1,2),(2,4),(3,8),(4,16) are shown. No connecting segments are part of the sequence.
Supplied Graph D shows the sequence d_n=2^n for integer indices n≥1, with its first four points displayed separately.
Index n
d_n
1
2
2
4
3
8
4
16
Write the recursive version d_1=2 and d_(n+1)=2d_n. Count the number of doublings needed to reach each displayed point. Leave the spaces between indices unconnected when describing the sequence itself.
Understand
For the power formula below, take r nonzero. If r=0, the initial term is A and every later term is zero; this avoids using an undefined 0⁰ expression.
An arithmetic sequence has a constant difference d between consecutive terms. Starting at index n₀ with value A, its recursive rule is a_(n+1)=a_n+d and its explicit rule is a_n=A+d(n−n₀). The factor n−n₀ counts how many changes occurred. Subtracting the explicit values at n+1 and n recovers the recursive difference, so the translation works in both directions.
A geometric sequence has a constant ratio r: a_(n+1)=r a_n. Starting with A at n₀ gives a_n=A r^(n−n₀). Dividing successive nonzero explicit terms recovers r. When A=0, every term is zero and a ratio cannot be inferred from the data by division; different r values produce the same zero sequence. If r is negative, signs alternate, which can be mathematically valid even when a physical population context rules it out.
A context determines which indices make sense. Monthly balances may use n=0,1,2,…, with zero denoting the starting observation. A machine’s completed-batch counts also use integers. If a subtractive arithmetic model eventually predicts negative inventory, restrict it to the feasible period or change the rule when inventory reaches zero. The algebraic expression can continue beyond that point, but the physical interpretation cannot silently do so.
A recursive rule emphasizes the mechanism between successive observations, while an explicit rule helps find a distant term without calculating every previous term. Deriving one from the other is a check on indexing. Substituting the starting index into the explicit formula should recover the stated initial value before any other calculations are trusted.
Apply
Compare an additive and multiplicative plan that have the same first-step increase. Give each in recursive and explicit form, then compare a later common index. Draw only the allowed discrete points or provide their coordinates. If you choose a continuous interpolation for planning, name it as a separate model and explain what its fractional input means.
Continue
Indexing is part of a sequence, not just notation. The next lesson changes the unit of continuous or discrete time in exponential models. Equivalent expressions must make the same predictions at the same physical time even when their numerical time inputs differ.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
An arithmetic sequence starts at a_1=7 and adds 4 each step. Find a_6.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A geometric sequence has b_0=96 and b_(n+1)=b_n/2. Find b_4.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Translate a_n=30−3(n−1), n≥1, into a recursive rule; then derive the explicit rule back from that recursion.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A geometric sequence begins at index 3 with c_3=5 and doubles per step. Give recursive and explicit rules and compute c_7.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Supplied Graph D plots sequence values 2,4,8,16 at indices 1,2,3,4. Explain why the graph has separate points and whether a value at index 2.5 is defined.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A fund begins with 200 dollars at index 0. Model adding 20 dollars after each month versus multiplying the balance by 1.1 after each month. Give both recursive and explicit models, and compare at month 3.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
