Ch 08 - Sequences and Series11th Mathematics
Strict Distinction: Sequence vs. Series
What is the strict mathematical distinction between a 'Sequence' and a 'Series'?
What is the strict mathematical distinction between a 'Sequence' and a 'Series'?
Sequence
An ordered list of numbers following a specific mathematical pattern (e.g., ). The strict positional order of elements matters.
Series
The mathematical arithmetic sum of the elements of a given sequence (e.g., ).
Prism Memory Hook
Sequence is the List. Series is the Sum.
Fibonacci Sequence Recurrence Relation
Fibonacci Sequence Recurrence Relation
The recurrence relation that formally defines the generation of the Fibonacci sequence.
Derivation
The sequence is initialized with the seed values and . Every subsequent term is mathematically generated by taking the arithmetic sum of the two immediately preceding terms.
Component Anatomy
- : The base initial boundary terms, both strictly defined as equal to .
- : The -th term of the sequence.
Boundary Conditions / Domain
- Valid strictly for integer positions .
Prism Memory Hook
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General Term of an Arithmetic Progression (A.P.)
General Term of an Arithmetic Progression (A.P.)
Standard formula for the general n-th term (a_n) of an Arithmetic Progression.
Derivation
By definition, the first term is . The second term is , the third is . The coefficient of is always exactly one less than the positional index .
Component Anatomy
- : The absolute first term of the sequence.
- : The common difference between any two consecutive terms.
- : The positional integer index of the target term.
Boundary Conditions / Domain
- (The position index must be a strictly positive integer).
Prism Memory Hook
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Also in this chapter
39 more cards to study in Prism
- 4Sum of the First n Terms of an A.P. ($S_n$)
- 5Extracting the Individual n-th Term from a Sum Formula
- 6Equidistant Terms Property of an A.P.
- 7Linear Transformations of an Arithmetic Progression
- 8SOP: Assuming Symmetrical Unknown Terms in an A.P.
- 9Identifying an A.P. from a Quadratic Sum Formula
- 10SOP: Formulating the Common Sequence of Two Intersecting A.P.s
- 11SOP: Decreasing A.P. in Loan Balance Instalments
- 12Single Arithmetic Mean (A.M.) Between Two Numbers
- 13SOP: Inserting $n$ Arithmetic Means Between Two Bounds
- 14Theorem: The Algebraic Sum of $n$ Inserted Arithmetic Means
- 15General Term of a Geometric Progression (G.P.)
- 16Sum of First $n$ Terms of a G.P. ($r \neq 1$)
- 17Sum to Infinity of a Convergent G.P.
- 18Equidistant Terms Property of a G.P.
- 19Exponential Transformations of a Geometric Progression
- 20Sequence Structure: Term-by-Term Product of Two G.P.s
- 21SOP: Assuming Symmetrical Unknown Terms in a G.P.
- 22Logarithmic Transformation of a Geometric Progression
- 23Inequality Trap: Identifying Hidden Geometric Progressions
- 24Lagrange's G.P. Identity
- 25Single Geometric Mean (G.M.) Between Two Numbers
- 26SOP: Inserting n Geometric Means Between Two Bounds
- 27Theorem: The Product of n Inserted Geometric Means
- 28Relationship Among A.M., G.M., and H.M.
- 29The AM-GM Inequality (Arithmetic-Geometric Mean)
- 30Constructing a Quadratic Equation from A.M. and G.M.
- 31Roots of a Quadratic via A.M. and G.M.
- 32Formula: Ratio of Numbers from their A.M. to G.M. Ratio
- 33Special Series: Sum of the First n Natural Numbers
- 34Special Series: Sum of the Squares of the First n Natural Numbers
- 35Special Series: Sum of the Cubes of the First n Natural Numbers
- 36Shortcut: Sum of Products of Two Consecutive Natural Numbers
- 37SOP: Sum of Repeated Digit Series
- 38Concept: The Method of Differences ($V_n$ Method)
- 39SOP: Evaluating a Telescoping Series
- 40Formula: Sum of an Infinite Arithmetico-Geometric Progression (A.G.P.)
- 41Mathematical Definition: Harmonic Progression (H.P.)
- 42Single Harmonic Mean (H.M.) Between Two Numbers
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