Patterns in Mathematics – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash, tailored for Class 6 in Mathematics.
This one-pager compiles key formulas and equations from the Patterns in Mathematics chapter of Ganita Prakash. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
n(n + 1) / 2
This formula calculates the sum of the first n natural numbers, where n is the count of numbers. It's useful to find the total when adding sequential numbers.
3n^2 - 3n + 1
Formula for the n-th triangular number, where n is the position in the sequence. Triangular numbers can represent objects arranged in triangular patterns.
n^2
This represents the formula for the n-th square number. The square numbers arise when you arrange dots in a square formation, useful in geometry.
n^3
This formula calculates the n-th cube number, where n is the position. It represents the volume of a cube and is applicable in spatial contexts.
1 + 3 + 5 + ... + (2n - 1) = n^2
This equation states that the sum of the first n odd numbers equals n squared. It illustrates a beautiful numeric property of odd numbers.
2^n
This formula indicates the n-th power of two. It's useful in computing exponential growth, relevant in technology and computer science.
1 + 2 + ... + n = n(n + 1) / 2
This represents the total sum of counting numbers up to n, useful in arithmetic series and combinatorial problems.
x^2 - y^2 = (x + y)(x - y)
This is the difference of squares formula. It factors a squared term minus another squared term, common in algebraic manipulations.
S_n = a / (1 - r)
For a geometric series, S_n indicates the sum where a is the first term and r is the common ratio. It’s useful in finance and growth modeling.
C(n, r) = n! / [r!(n - r)!]
This binomial coefficient formula computes combinations of n items taken r at a time, vital in probability and statistics.
Equations
T_n = n(n + 1) / 2
T_n indicates the n-th triangular number, calculated as n times (n + 1) divided by 2. Used to solve problems involving triangular arrangements.
S = a + a(1 + r) + a(1 + r)^2 + ... + a(1 + r)^(n - 1)
This equation represents the sum of a geometric sequence, where S is the total, a is the first term, r is the common ratio, and n is the number of terms.
P(n) = n(n - 1) / 2
This represents the number of edges in a complete graph of n points, illustrating relationships in graph theory.
F_n = F_(n-1) + F_(n-2) (Fibonacci Sequence)
This recursive equation defines the Fibonacci Sequence, where each number is the sum of the two preceding ones. Common in nature and computer algorithms.
H_n = 1 + 1/2 + 1/3 + ... + 1/n
This represents the n-th Harmonic number, summing the reciprocals of the first n natural numbers, important in analysis and number theory.
L = 3x + 2y
This linear equation computes the value L based on variables x and y, often found in calculation of linear relationships.
m = (y2 - y1) / (x2 - x1)
This defines the slope (m) between two points (x1, y1) and (x2, y2), critical in graphing linear equations.
y = mx + b
The slope-intercept equation of a line, where m is the slope and b is the y-intercept. Essential for graphing linear relationships.
A = πr^2
This formula calculates the area (A) of a circle with radius r. It's used in geometry, particularly in problems involving circular areas.
V = s^3
This formula gives the volume (V) of a cube with side length s. It’s practical in volume measurement scenarios.
