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Moreover, since A n A m = A n+m for any square matrix A, the following identities can be derived (they are obtained from two different coefficients of the matrix product, and one may easily deduce the second one from the first one by changing n into n + 1), + = +, + + = +.
When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.
If a and b have different signs, define a + b to be the difference between |a| and |b|, with the sign of the term whose absolute value is larger. [61] As an example, −6 + 4 = −2; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.
For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative , there is no need for parentheses, and the result is the same irrespective of the order of the summands.
Here the first term represents the area of the blue triangle, the second term is the area of the two green triangles, the third term is the area of the four yellow triangles, and so on. Simplifying the fractions gives 1 + 1 4 + 1 16 + 1 64 + ⋯ , {\displaystyle 1+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{64}}+\cdots ,} a geometric series with ...
Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum is less than 1. A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270.
the exponents of y in the terms are 0, 1, 2, ..., n − 1, n (the first term implicitly contains y 0 = 1); the coefficients form the n th row of Pascal's triangle; before combining like terms, there are 2 n terms x i y j in the expansion (not shown); after combining like terms, there are n + 1 terms, and their coefficients sum to 2 n.
The Brahmagupta–Fibonacci identity states that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse, given n = a 2 + b 2 = c 2 + d 2 {\displaystyle n=a^{2}+b^{2}=c^{2}+d^{2}} we find n {\displaystyle n} as a product of sums of two squares.