The reasoning process of the formula of binomial theorem

The reasoning process of the formula of binomial theorem

(x+a)^n=∑_ (k = 0) ^ n &; ((n &; K) x ^ k a ^ (n-k)) hope to be adopted

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Binomial theorem, also known as Newton's binomial theorem, was put forward by Isaac Newton in 1664 and 1665. This theorem points out that the binomial coefficient refers to the polynomial on the right side of the equal sign, which is called binomial expansion. The general formula of binomial expansion is its i-term coefficient, which can be expressed as: see figure right, that is, n takes

Finding binomial theorem formula and difference product formula

Answer: quadratic term theorem
A + b) n power = C (n, 0) a (n power) + C (n, 1) a (n-1 power) B (1 power) + +C (n, R) a (N-R power) B (r power) + +C (n, n) B (nth power) (n ∈ n *)
C (n, 0) means to take 0 out of n,
This formula is called binomial theorem. The polynomial on the right is called the quadratic expansion of (a + b) n, where the coefficient CNR (r = 0,1,...) n) It is called the coefficient of quadratic term, cnran RBR in the formula. It is called the general term of binomial expansion, which is expressed by tr + 1, that is, the general term is the R + 1 term of the expansion: tr + 1 = cnraa RBR
It is shown that (1) tr + 1 = cnraa RBR is the R + 1 term of the expansion of (a + b) n. r = 0,1,2 n. It is different from cnrbn rar, the R + 1 term of the expansion of (B + a) n
② TR + 1 only refers to the standard form of (a + b) n. The general formula of binomial expansion of (a-b) n is tr + 1 = (- 1) rcnran RBR
③ The coefficient CNR is called the binomial coefficient of the R + 1 degree of the expansion, which should be distinguished from the coefficient of the R + 1 term with respect to one or more letters
In particular, in the binomial theorem, if a = 1, B = x, then we get the formula:
(1+x)n＝1+cn1x+Cn2x2+… +Cnrxa+… +xn.
When n is a small positive integer, we can use Yang Hui triangle to write the phase
The formula of product sum difference is as follows
sinαsinβ=-[cos(α+β)-cos(α-β)]/2
cosαcosβ=[cos(α+β)+cos(α-β)]/2
sinαcosβ=[sin(α+β)+sin(α-β)]/2
cosαsinβ=[sin(α+β)-sin(α-β)]/2
Sum difference product formula:
sinθ+sinφ=2sin[(θ+φ)/2]cos[(θ-φ)/2]
sinθ-sinφ=2cos[(θ+φ)/2]sin[(θ-φ)/2]
cosθ+cosφ=2cos[(θ+φ)/2]cos[(θ-φ)/2]
cosθ-cosφ=-2sin[(θ+φ)/2]sin[(θ-φ)/2](X-Y)]