š Key Concepts
Sequences and their termsArithmetic Progression (AP)- Common difference
- nth term
- Sum of n terms
Geometric Progression (GP)- Common ratio
- nth term
- Sum of n terms
Arithmetic Mean (AM)Geometric Mean (GM)AM-GM inequalityRelationship between AM and GMā
šÆ Key Formulas
AP:- nth term (aā) = a + (n-1)d
- Sum (Sā) = n/2[2a + (n-1)d] = n/2(a + l)
GP:- nth term (aā) = ar^(n-1)
- Sum (Sā) = a(1-r^n)/(1-r) [r ā 1]
- Sum (Sā) = a(r^n-1)/(r-1) [r > 1]
AM = (aā + aā + ... + aā)/nGM = āæā(aā Ć aā Ć ... Ć aā)AM ā„ GM for positive real numbersā
ā ļø Common Mistakes to Avoid
Wrong common difference/ratio identificationIncorrect application of sum formulasConfusion between AP and GPWrong application of AM-GM inequalityErrors in term calculationsForgetting conditions for sum of GPā
š Knowledge Prerequisites
Basic algebraNumber systemExponents and powersBasic inequalitiesā
š” Tips for Students
Verify sequence type firstCheck first few terms to confirm patternRemember AM ā„ GM alwaysUse common difference/ratio to identify progressionPractice finding missing termsā
š Practice Recommedations
Find terms in sequencesCalculate sums of seriesSolve for missing termsApply AM-GM inequalityWork with arithmetic meansPractice geometric meansCombine AP and GP conceptsā
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