Study guide: Algebra2: Sequences, Sums, and Recursion
A sequence distinguishes an indexed term from the sum of several terms. Arithmetic and geometric structures support different formulas, while a recursive definition determines each next term from previously defined values.
Ideas to explain in your own words
- An arithmetic sequence adds the same common difference between consecutive terms. Its nth term equals the first value plus n−1 copies of that difference because index one starts before any step. The number of steps is one less than the term position.
- A geometric sum adds several values whose consecutive ratio is fixed. Explicitly listing a short supplied sequence distinguishes its total from its last term and from its number of terms. A geometric term formula alone does not give a sum.
- Terms equally far from opposite ends of an arithmetic sequence have equal pair sums. Pairing gives the sum formula as number of terms times the average of first and last. Use the stated number of terms, including both endpoints.
- A recursive definition names a starting term and an operation producing the next term. Substituting the previous term’s actual value follows that definition one step at a time. The subscript labels a position rather than multiplication.
Useful vocabulary
- arithmetic sequence
- Consecutive terms have a constant difference.
- finite series
- A sum of a stated finite set of sequence terms.
- recursion
- Defining a term using previous terms and initial values.
Watch your reasoning
- The number of steps is one less than the term position. Do not use a geometric exponent for an additive sequence; write early terms to verify indexing and the stated common difference.
- A geometric term formula alone does not give a sum. Check whether the question asks for one term or cumulative total; for a finite sum formula, the special case ratio one needs separate handling.
- Use the stated number of terms, including both endpoints. The average of endpoints equals the mean for an arithmetic sequence, but that shortcut need not work for an arbitrary list.
A worked review example
Complete supplied guide examples: Count Equal Additive Steps from the First Term — An arithmetic sequence has a₁=2, common difference 3. Find a_3. Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9. Term 3 is 2 steps after term 1. Therefore a_3=2+3(2)=8. The exponent rule for geometric sequences is not used. Add Terms Rather than Report the Last Term — Find the sum of the first three terms of a geometric sequence starting at 2 with ratio 2. Candidate answers: (A) 13 (B) 14 (C) 15 (D) 28. The three terms are 2,4,8. Their sum is 14. Using a(1−r³)/(1−r) gives the same result when r=2. Pair First and Last Terms in an Arithmetic Sum — Find the sum of the four-term arithmetic sequence 2,4,6,8. Candidate answers: (A) 19 (B) 20 (C) 21 (D) 40. Pair first with last and second with third; each pair sums to 10. Two pairs total 20. Equivalently S=4(first+last)/2. Apply a Recursive Rule to the Previous Value — A recursive sequence has a₁=2 and a_(n+1)=3a_n+1. Find a₂. Candidate answers: (A) 14 (B) 6 (C) 7 (D) 8. Apply the supplied rule once to a₁: a₂=3(2)+1=7. The subscript n marks a term position, not a multiplication by n.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.