difference between arithmetic sequence ang harmonic sequence
Answer (1)
Answer:Here's a breakdown of the differences between arithmetic sequences and harmonic sequences:Arithmetic SequenceDefinition: A sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference.Formula: a<sub>n</sub> = a<sub>1</sub> + (n - 1)d, where:a<sub>n</sub> is the nth terma<sub>1</sub> is the first termd is the common differencen is the position of the term in the sequenceExample: 2, 5, 8, 11, 14... (common difference = 3)Harmonic SequenceDefinition: A sequence where the reciprocals of the terms form an arithmetic sequence.Formula: There is no single formula for a harmonic sequence. Instead, the reciprocals of the terms follow the arithmetic sequence formula.Example: 1, 1/2, 1/3, 1/4, 1/5... (The reciprocals form the arithmetic sequence 1, 2, 3, 4, 5...)Key Differences:Feature Arithmetic Sequence Harmonic SequencePattern Constant difference between consecutive terms Reciprocals of terms form an arithmetic sequenceFormula a<sub>n</sub> = a<sub>1</sub> + (n - 1)d No single formula, but reciprocals follow arithmetic sequence formulaExample 2, 5, 8, 11... 1, 1/2, 1/3, 1/4...In summary:Arithmetic sequences have a constant difference between terms, while harmonic sequences have a constant difference between the reciprocals of their terms.Arithmetic sequences have a simple formula to calculate any term, while harmonic sequences rely on the arithmetic formula for their reciprocals.Applications:Arithmetic Sequences: Used in finance, time management, sports statistics, and other fields where regular increments or decrements occur.Harmonic Sequences: Used in physics, music theory, and other fields dealing with inverse relationships.Step-by-step explanation:
