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Questions and Answers
What is the recursive formula for an arithmetic sequence?
What is the recursive formula for an arithmetic sequence?
- f(n) = f(n-1) * r
- a1(1-r^n)/(1-r)
- f(n) = f(n-1) + d (correct)
- n(a1 + an)/2
What is the recursive formula for a geometric sequence?
What is the recursive formula for a geometric sequence?
- an=a1(r)^(n-1)
- f(n) = f(n-1) * r (correct)
- a1+(n-1)d
- f(n) = f(n-1) + d
What is the formula for the arithmetic sum?
What is the formula for the arithmetic sum?
n(a1 + an)/2
What is the formula for the geometric sum?
What is the formula for the geometric sum?
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What is the formula for an arithmetic sequence?
What is the formula for an arithmetic sequence?
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What is the formula for a geometric sequence?
What is the formula for a geometric sequence?
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How do you find the common difference in an arithmetic sequence?
How do you find the common difference in an arithmetic sequence?
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How do you find the common ratio in a geometric sequence?
How do you find the common ratio in a geometric sequence?
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Study Notes
Arithmetic and Geometric Sequences
- Arithmetic Sequence Recursive Formula: Defined as ( f(n) = f(n-1) + d ) or ( A_n = A_{n-1} + d ), where ( d ) is the common difference between terms.
- Geometric Sequence Recursive Formula: Expressed as ( f(n) = f(n-1) \times r ) or ( A_n = A_{n-1} \times r ), where ( r ) is the common ratio.
Sums of Sequences
- Arithmetic Sum Formula: Calculated using ( S_n = \frac{n(a_1 + a_n)}{2} ), where ( S_n ) is the sum of the first ( n ) terms, ( a_1 ) is the first term, and ( a_n ) is the last term.
- Geometric Sum Formula: Given by ( S_n = \frac{a_1(1 - r^n)}{1 - r} ) for ( r \neq 1 ), where ( S_n ) represents the sum of the first ( n ) terms of a geometric series.
Formulas for General Terms
- Arithmetic Sequence General Formula: Expressed as ( a_n = a_1 + (n-1)d ), indicating the ( n^{th} ) term can be found by adding the first term to ( (n-1) ) times the common difference.
- Geometric Sequence General Formula: Written as ( a_n = a_1(r)^{n-1} ), which allows determination of the ( n^{th} ) term by using the first term multiplied by the common ratio raised to the power of ( (n-1) ).
Finding Sequence Properties
- Common Difference: Determined by the formula ( \frac{\text{value}_y - \text{value}_x}{\text{term}_y - \text{term}_x} ), which calculates how much each term increases or decreases.
- Common Ratio: Found using ( \frac{\text{term}_y}{\text{term}_x} ) for ( \sqrt{x \text{ value} / y \text{ value}} ), indicating the factor by which each successive term is multiplied.
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