Formula Sheet: Sequences and Series
This chapter discusses sequences, which are ordered lists of numbers, and their importance in mathematics. It covers different types of sequences and series, including arithmetic and geometric progressions, and their applications.
Sequences and Series – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Sequences and Series chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formula sheet
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
General term of a sequence: a_n
a_n is the value of the nth term of a sequence. It allows for determining specific terms based on a given rule or formula.
Arithmetic Progression (A.P.): a_n = a + (n-1)d
a represents the first term, d is the common difference, and n represents the term number. Used to find the value of any term in an A.P.
Sum of first n terms of A.P.: S_n = n/2 [2a + (n - 1)d]
S_n is the sum of the first n terms. Utilized for calculating total values in arithmetic sequences.
Geometric Progression (G.P.): a_n = ar^(n-1)
a is the first term, r is the common ratio, and n is the term number. This formula helps find specific terms in a geometric sequence.
Sum of first n terms of a G.P.: S_n = a(1 - r^n) / (1 - r) (if r ≠ 1)
S_n indicates the sum of the first n terms of the G.P. Essential in financial calculations involving growth.
Fibonacci Sequence: a_n = a_(n-1) + a_(n-2)
Defines each term as the sum of the two preceding terms. Widely used in algorithms and natural phenomena.
Relationship between A.M. and G.M.: A ≥ G
A is the Arithmetic Mean, G is the Geometric Mean. Indicates that the Arithmetic Mean is always greater than or equal to the Geometric Mean.
Sum of squares of first n natural numbers: S = n(n + 1)(2n + 1) / 6
Provides a way to calculate the sum of squares, useful in statistical formulas.
Sum of cubes of first n natural numbers: S = (n(n + 1) / 2)²
This formula allows for the calculation of the sum of cubes, useful in advanced mathematical applications.
Arithmetic Mean: A = (a + b) / 2
Calculates the average of two numbers, serving as a fundamental concept in statistics.
Equations
2nd term of A.P.: a_2 = a + d
Indicates the value of the second term in an A.P., where d is the common difference.
Common ratio in G.P.: r = a_2 / a_1
Establishes the relationship between two consecutive terms of a G.P.
S_n = n/2 (first term + last term)
Another method to calculate sum of first n terms, useful when the first and last terms are known.
a = (a_1 * a_n)^(1/n)
Hypothetical formula component relating the means in geometric sequences.
n = log(a_n/a) / log(r)
Determines the term number n in a G.P. when a term value a_n and the first term a with the common ratio r are known.
Cyclic sum of A.M. and G.M.: A - G = (A^2 - G^2) / (A + G)
Shows the relationship between the means derived from the same set of values.
Sum of n terms via nth term: S_n = n/2 [2a + d(n - 1)]
An alternative to find sum when n and d are known.
G.M. of n positive terms: G = (x_1 * x_2 * ... * x_n)^(1/n)
Calculates geometric mean and serves as an efficient mean calculation tool.
n = (1 - r^N) / (1 - r)
Specific derivation form in calculating terms' sums in G.P.
Sum of infinite G.P.: S = a / (1 - r) (if |r| < 1)
Calculates the sum of an infinite geometric series, pivotal in series convergence discussions.
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