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Concept · Functions & Sequences
Arithmetic and geometric sequences
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A Functions & Sequences set, untimed
Named traps
9 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Arithmetic/geometric formula mixedUses the arithmetic term/sum formula for a geometric sequence or vice versa.→
- Common ratio misidentifiedDerives the wrong common ratio from a given term.→
- Extremal-sum index off by oneLocates the maximizing/last-positive index off by one.→
- Geometric monotonicity q-sign ignoredIgnores negative-ratio or q-in-(0,1) vs q>1 cases when judging monotonicity sufficiency/necessity.→
- Geometric-sum offset not cancelledForgets that Sn=a·bn+c requires a+c=0 for a true geometric series.→
- Ratio exponent off by oneUses n instead of n-1 (or similar) in the geometric general-term exponent.→
- Sn treated as linearTreats the arithmetic partial sum as linear rather than quadratic in n when maximizing.→
- Symmetric-sum pairing misusedMisuses Sn=n(a1+an)/2 pairing when reasoning about sign or extremal partial sums.→
- Zero-crossing sign reasoning wrongDraws the wrong sign conclusion for terms around where the sequence crosses zero.→
Key formulas
7- Arithmetic nth terman = a1 + (n − 1)d
- Arithmetic sumSn = n(a1 + an)/2 = n·a1 + n(n−1)d/2 (quadratic in n)
- Geometric nth terman = a1·rn−1
- Geometric sum (r ≠ 1): Sn = a1(1 − rn)/(1 − r)
- Geometric mean of a,bG = ±√ab
- Term from partial suman = Sn − Sn−1 for n ≥ 2, a1 = S1
- Geometric-series partial-sum form Sn = A·rn + C requires A + C = 0
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeAlgebra 2 foundations