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107 results for “sum”

Formulae (103)

Average (Mean)

\text{average} = \frac{\text{sum of values}}{\text{number of values}}

Add all values and divide by how many there are.

Arithmetic & BasicsBasic

Place Value

N = \sum d_i \times 10^{i}

Each digit’s value depends on its position.

Arithmetic & BasicsBasic

Divisibility by 3

\text{digit sum divisible by 3}

A number is divisible by 3 if its digit sum is.

Arithmetic & BasicsBasic

Divisibility by 9

\text{digit sum divisible by 9}

A number is divisible by 9 if its digit sum is.

AlgebraIntermediate

Binomial Theorem

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expands any power of a binomial into a sum of terms.

AlgebraIntermediate

Logarithm Product Rule

\log_b(xy) = \log_b(x) + \log_b(y)

The logarithm of a product equals the sum of logarithms.

AlgebraIntermediate

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factors the sum of two cubes.

AlgebraBasic

Arithmetic Series Sum

S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence.

AlgebraIntermediate

Geometric Series Sum

S_n = a_1 \frac{1 - r^n}{1 - r}

Sum of the first n terms of a geometric sequence (r ≠ 1).

AlgebraIntermediate

Infinite Geometric Series

S = \frac{a_1}{1 - r}, \quad |r| < 1

Sum of an infinite geometric series that converges.

AlgebraAdvanced

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

Closed form for the sum of the first n cubes.

AlgebraIntermediate

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factoring formula for a sum of two cubes.

AlgebraIntermediate

Logarithm Product Rule

\log_b(xy) = \log_b x + \log_b y

The log of a product is the sum of the logs.

AlgebraAdvanced

Harmonic Mean

HM = \frac{n}{\sum 1/x_i}

Reciprocal of the average of reciprocals.

AlgebraAdvanced

Infinite Geometric Series

S = \frac{a}{1-r}, \; |r|<1

Sum of an infinite geometric series that converges.

AlgebraIntermediate

Finite Geometric Series

S_n = a\frac{1-r^n}{1-r}

Sum of the first n terms of a geometric series.

Geometry & TrigonometryIntermediate

Sine Angle Sum

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

Angle addition formula for sine.

Geometry & TrigonometryIntermediate

Cosine Angle Sum

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

Angle addition formula for cosine.

Geometry & TrigonometryBasic

Perimeter of a Triangle

P = a + b + c

Sum of the three sides of a triangle.

Geometry & TrigonometryBasic

Perimeter of a Trapezoid

P = a + b + c + d

Sum of all four sides of a trapezoid.

Geometry & TrigonometryBasic

Sum of Interior Angles

S = (n - 2)\times 180^\circ

Total of interior angles of an n-sided polygon.

Geometry & TrigonometryBasic

Interior Angle Sum

S = (n-2)\times 180\degree

Sum of interior angles of an n-sided polygon.

TrigonometryIntermediate

Sine of a Sum

\sin(A+B) = \sin A\cos B + \cos A\sin B

Angle addition formula for sine.

TrigonometryIntermediate

Cosine of a Sum

\cos(A+B) = \cos A\cos B - \sin A\sin B

Angle addition formula for cosine.

TrigonometryAdvanced

Tangent of a Sum

\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

Angle addition formula for tangent.

TrigonometryAdvanced

Product: sin A cos B

\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

Product-to-sum identity.

TrigonometryAdvanced

Product: cos A cos B

\cos A\cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]

Product-to-sum identity.

TrigonometryAdvanced

Product: sin A sin B

\sin A\sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

Product-to-sum identity.

TrigonometryAdvanced

Sum to Product: sin A + sin B

\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: sin A − sin B

\sin A - \sin B = 2\cos\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: cos A + cos B

\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: cos A − cos B

\cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

CalculusAdvanced

Taylor Series

f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n

Expands a function as an infinite polynomial around a.

CalculusAdvanced

Maclaurin Series for e^x

e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}

Taylor series of the exponential at 0.

CalculusBasic

Sum Rule (Derivative)

(f+g)' = f' + g'

The derivative of a sum is the sum of the derivatives.

Statistics & ProbabilityBasic

Arithmetic Mean

\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i

The average of a data set.

Statistics & ProbabilityIntermediate

Variance

\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2

Average squared deviation from the mean.

Statistics & ProbabilityIntermediate

Standard Deviation

\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2}

Square root of the variance; typical spread.

Statistics & ProbabilityIntermediate

Expected Value

E[X] = \sum_i x_i\,p_i

The long-run average value of a random variable.

Statistics & ProbabilityAdvanced

Pearson Correlation

r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}}

Measures linear correlation between two variables.

Statistics & ProbabilityAdvanced

Covariance

\text{cov}(X,Y) = \frac{1}{n}\sum (x_i-\bar{x})(y_i-\bar{y})

Measures how two variables vary together.

Statistics & ProbabilityAdvanced

Chi-Square Statistic

\chi^2 = \sum \frac{(O-E)^2}{E}

Tests goodness of fit between observed and expected counts.

Statistics & ProbabilityIntermediate

Weighted Mean

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

Average where values carry different weights.

Statistics & ProbabilityIntermediate

Sample Variance

s^2 = \frac{1}{n-1}\sum(x_i-\bar{x})^2

Unbiased estimate of population variance.

Probability & CombinatoricsIntermediate

Expected Value

E[X] = \sum_{i} x_i\, p_i

Long-run average value of a random variable.

Data Science & MLBasic

Mean Squared Error (MSE)

MSE = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2

Average of squared differences between predicted and actual values.

Data Science & MLBasic

Root Mean Squared Error (RMSE)

RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}

Square root of MSE, in the same units as the target variable.

Data Science & MLBasic

Mean Absolute Error (MAE)

MAE = \frac{1}{n}\sum_{i=1}^{n}|y_i - \hat{y}_i|

Average of absolute differences between predictions and actual values.

Data Science & MLIntermediate

Coefficient of Determination (R²)

R^2 = 1 - \frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}

Proportion of variance in the target explained by the model.

Data Science & MLAdvanced

Softmax Function

\sigma(z_i) = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}

Converts a vector of scores into a probability distribution.

Data Science & MLAdvanced

Binary Cross-Entropy Loss

L = -\frac{1}{n}\sum[y\log(\hat{y}) + (1-y)\log(1-\hat{y})]

Loss function for binary classification.

Data Science & MLBasic

Euclidean Distance

d = \sqrt{\sum_{i=1}^{n}(a_i - b_i)^2}

Straight-line distance between two points in n-dimensional space.

Data Science & MLBasic

Manhattan Distance

d = \sum_{i=1}^{n}|a_i - b_i|

Distance measured along axes at right angles (taxicab geometry).

Data Science & MLAdvanced

Shannon Entropy

H = -\sum_{i=1}^{n} p_i \log_2 p_i

Measures the uncertainty or information content of a distribution.

Data Science & MLIntermediate

Gini Impurity

Gini = 1 - \sum_{i=1}^{n} p_i^2

Probability of misclassifying a random element; used in decision trees.

Data Science & MLAdvanced

Categorical Cross-Entropy

L = -\sum_{i=1}^{K} y_i \log(\hat{y}_i)

Loss for multi-class classification with one-hot labels.

Linear AlgebraIntermediate

Dot Product

\vec{a}\cdot\vec{b} = \sum_i a_i b_i = |a||b|\cos\theta

Scalar product of two vectors.

Linear AlgebraIntermediate

Trace of a Matrix

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Linear AlgebraIntermediate

Matrix Trace

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

Linear AlgebraBasic

Dot Product

\mathbf{a}\cdot\mathbf{b} = \sum a_i b_i

Sum of componentwise products of two vectors.

Linear AlgebraBasic

Vector Magnitude

\|\mathbf{v}\| = \sqrt{\sum v_i^2}

Length of a vector.

Physics Laws & TheoremsBasic

Newton's First Law (Inertia)

\sum F = 0 \Rightarrow v = \text{const}

An object stays at rest or in uniform motion unless acted on by a net force.

Physics Laws & TheoremsIntermediate

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum of an isolated system stays constant.

Physics Laws & TheoremsIntermediate

Kirchhoff's Current Law

\sum I_{in} = \sum I_{out}

Current into a junction equals current out of it (charge conservation).

Physics Laws & TheoremsIntermediate

Kirchhoff's Voltage Law

\sum V = 0

The sum of voltages around any closed loop is zero.

Physics Laws & TheoremsBasic

Newton First Law

\sum F = 0 \Rightarrow \text{constant } v

An object stays at rest or in uniform motion unless acted upon.

Physics Laws & TheoremsIntermediate

Conservation of Momentum

\sum p_i = \sum p_f

Total momentum is conserved without external forces.

Physics Laws & TheoremsIntermediate

Conservation of Charge

\sum q = \text{constant}

Total electric charge in an isolated system is conserved.

Electronics & CircuitsIntermediate

Kirchhoff's Voltage Law

\sum V = 0

Voltages around any closed loop sum to zero.

Electronics & CircuitsIntermediate

Kirchhoff's Current Law

\sum I_{in} = \sum I_{out}

Current into a node equals current out.

ChemistryAdvanced

Hess Law

\Delta H = \sum \Delta H_{steps}

Total enthalpy change is the sum of step changes.

Biology & Life SciencesBasic

Allele Frequency Sum

p + q = 1

The two allele frequencies must total 1.

Biology & Life SciencesAdvanced

Shannon Diversity Index

H = -\sum p_i \ln p_i

Measures species diversity in a community.

Biology & Life SciencesAdvanced

Simpson Diversity Index

D = 1 - \sum p_i^2

Probability two random individuals differ in species.

Biology & Life SciencesIntermediate

Respiratory Quotient

RQ = \frac{CO_2\ produced}{O_2\ consumed}

Ratio revealing which fuel is being metabolised.

Biology & Life SciencesAdvanced

Biochemical Oxygen Demand

BOD = \frac{D_1 - D_2}{P}

Oxygen consumed by microbes decomposing organic matter.

Biology & Life SciencesBasic

Allele Frequency

p + q = 1

Frequencies of two alleles sum to one.

Biology & Life SciencesAdvanced

Respiratory Quotient

RQ = \frac{CO_2\text{ produced}}{O_2\text{ consumed}}

Ratio revealing which fuel is being metabolized.

Health & FitnessIntermediate

MET to Oxygen Uptake

VO_2 = 3.5 \times MET

Oxygen consumption from a MET activity level.

Aviation & AeronauticsAdvanced

Takeoff Ground Roll

s = \dfrac{V_{LO}^2}{2a}

Approximate ground-roll distance assuming constant acceleration to lift-off speed.

Finance & InterestBasic

Future Value

FV = PV(1 + r)^n

Future worth of a present sum.

Finance & InterestBasic

Present Value

PV = \frac{FV}{(1 + r)^n}

Present worth of a future sum.

Finance & InterestAdvanced

Net Present Value

NPV = \sum_{t=0}^{n}\frac{C_t}{(1+r)^t}

Present value of a stream of cash flows.

EconomicsIntermediate

GDP Deflator

Deflator = \dfrac{Nominal\ GDP}{Real\ GDP}\times 100

A broad price index covering all goods in GDP, not just a consumer basket.

EconomicsIntermediate

Marginal Propensity to Consume

MPC = \dfrac{\Delta C}{\Delta Y}

The fraction of an extra dollar of income that is spent rather than saved.

EconomicsAdvanced

Consumer Surplus

CS = \tfrac{1}{2}\times \Delta P \times \Delta Q

The benefit consumers gain when they pay less than the maximum they were willing to pay.

EconomicsIntermediate

Consumer Price Index

CPI = \frac{\text{Basket now}}{\text{Basket base}}\times 100

Measures price levels relative to a base year.

EconomicsAdvanced

Spending Multiplier

k = \frac{1}{1 - MPC}

Total impact of spending given marginal consumption.

Environmental ScienceBasic

Energy Consumption

E = P \times t

Electrical energy used over time.

Environmental ScienceBasic

Water Use Intensity

WUI = \frac{V_{water}}{Output}

Water consumed per unit of product or service.

Number TheoryBasic

Sum of First n Integers

\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

Closed form for the sum of the first n natural numbers.

Number TheoryIntermediate

Sum of First n Squares

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Closed form for the sum of the first n squares.

Number TheoryBasic

Divisibility Rule for 3

n \text{ divisible by } 3 \iff \text{digit sum divisible by } 3

A number is divisible by 3 if the sum of its digits is.

Number TheoryBasic

Divisibility Rule for 9

n \text{ divisible by } 9 \iff \text{digit sum divisible by } 9

A number is divisible by 9 if the sum of its digits is.

Number TheoryIntermediate

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

The sum of the first n cubes equals the square of the sum.

Number TheoryAdvanced

Sum of Divisors

\sigma(n) = \prod \frac{p_i^{a_i+1}-1}{p_i - 1}

Sum of all divisors from prime factorization.

Number TheoryBasic

Triangular Number

T_n = \frac{n(n+1)}{2}

Sum of the first n natural numbers.

Number TheoryBasic

Fibonacci Recurrence

F_n = F_{n-1} + F_{n-2}

Each term is the sum of the two before it.

Computer ScienceAdvanced

Shannon Entropy

H = -\sum_i p_i \log_2 p_i

Average information content of a random source (bits).

Computer ScienceAdvanced

Hamming Distance

d(x,y) = \sum_i [x_i \ne y_i]

Number of positions at which two strings differ.

Computer ScienceBasic

Binary to Decimal

N = \sum_{i=0}^{k} b_i 2^i

Converts a binary number to decimal.

Computer ScienceBasic

Hexadecimal to Decimal

N = \sum_{i=0}^{k} h_i 16^i

Convert a hexadecimal number to decimal.

Computer ScienceBasic

Hexadecimal to Decimal

d = \sum h_i \times 16^i

Convert a hex number to decimal by positional weights.

Math Tricks (4)

Casting Out Nines (Verification)

A quick way to verify multiplication and addition results by checking digit sums modulo 9.

View trick

Duplex Method for Squaring

The Dwandwa (duplex) method squares any number by summing duplex values digit by digit — a general Vedic squaring technique.

View trick

Sum of 1 to N Instantly

Gauss's formula: the sum of the first n whole numbers is n(n+1)/2 — no adding required.

View trick

Sum of First N Odd Numbers = N²

The sum of the first n odd numbers is always a perfect square, exactly n².

View trick