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Essential Identities/Ideas

40 of 40 entries listed.

  1. 1.Sum of the first n integers
    1+2+⋯+n=n(n+1)21+2+\cdots+n=\frac{n(n+1)}{2}
  2. 2.Sum of squares
    12+22+⋯+n2=n(n+1)(2n+1)61^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}
  3. 3.Sum of cubes
    13+23+⋯+n3=(n(n+1)2)21^3+2^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2
  4. 4.Geometric series
    1+r+r2+⋯+rn=rn+1−1r−11+r+r^2+\cdots+r^n=\frac{r^{n+1}-1}{r-1}
    ∣r∣<1:1+r+r2+⋯=11−r|r|<1:\quad 1+r+r^2+\cdots=\frac{1}{1-r}
  5. 5.Telescoping series
    11⋅2+12⋅3+⋯+1n(n+1)=(1−12)+⋯+(1n−1n+1)=1−1n+1\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{n(n+1)}=\left(1-\frac{1}{2}\right)+\cdots+\left(\frac{1}{n}-\frac{1}{n+1}\right)=1-\frac{1}{n+1}
  6. 6.Square of a sum
    (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2
  7. 7.Square of a difference
    (a−b)2=a2−2ab+b2(a-b)^2=a^2-2ab+b^2
  8. 8.Cube of a sum
    (a+b)3=a3+3a2b+3ab2+b3(a+b)^3=a^3+3a^2b+3ab^2+b^3
  9. 9.Cube of a difference
    (a−b)3=a3−3a2b+3ab2−b3(a-b)^3=a^3-3a^2b+3ab^2-b^3
  10. 10.Difference of squares
    a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)
  11. 11.Difference of cubes
    a3−b3=(a−b)(a2+ab+b2)a^3-b^3=(a-b)(a^2+ab+b^2)
  12. 12.Sum of cubes (factored)
    a3+b3=(a+b)(a2−ab+b2)a^3+b^3=(a+b)(a^2-ab+b^2)
  13. 13.General difference of powers
    an−bn=(a−b)(an−1+an−2b+⋯+bn−1)a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+\cdots+b^{n-1}\right)
  14. 14.Sum of odd power
    n odd:an+bn=(a+b)(an−1−an−2b+⋯+bn−1)n\text{ odd}:\quad a^n+b^n=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots+b^{n-1}\right)
  15. 15.Difference of even power
    n even:an−bn=(a+b)(an−1−an−2b+⋯−bn−1)n\text{ even}:\quad a^n-b^n=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots-b^{n-1}\right)
  16. 16.Hockey Stick Identity
    (kk)+(k+1k)+(k+2k)+⋯+(nk)=(n+1k+1)\binom{k}{k}+\binom{k+1}{k}+\binom{k+2}{k}+\cdots+\binom{n}{k}=\binom{n+1}{k+1}
  17. 17.Vandermonde's Identity
    (m0)(nk)+(m1)(nk−1)+(m2)(nk−2)+⋯+(mk)(n0)=(m+nk)\binom{m}{0}\binom{n}{k}+\binom{m}{1}\binom{n}{k-1}+\binom{m}{2}\binom{n}{k-2}+\cdots+\binom{m}{k}\binom{n}{0}=\binom{m+n}{k}
  18. 18.Symmetric cubic sum
    a3+b3+c3=(a+b+c)(a2+b2+c2−ab−ac−bc)+3abca^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc
  19. 19.Factoring quadratic by grouping
    x2+(a+b)x+ab=(x+a)(x+b)x^2+(a+b)x+ab=(x+a)(x+b)
  20. 20.Binomial Theorem
    (a+b)n=(n0)an+(n1)an−1b+(n2)an−2b2+⋯+(nn)bn(a+b)^n=\binom{n}{0}a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+\cdots+\binom{n}{n}b^n
  21. 21.Quadratic Formula
    ax2+bx+c=0  ⇒  x=−b±b2−4ac2aax^2+bx+c=0\;\Rightarrow\;x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
  22. 22.Approaches to Similar Equations
    Addition and Subtraction.\text{Addition and Subtraction.}
  23. 23.SFFT (Simon's Favorite Factoring Trick)
    xy+ax+by=c  ⇒  (x+b)(y+a)=c+abxy+ax+by=c\;\Rightarrow\;(x+b)(y+a)=c+ab
    GSFFT (General Form)\text{GSFFT (General Form)}
    axy+bx+cy=daxy+bx+cy=d
    a2xy+abx+acy=ada^2xy+abx+acy=ad
    (ax+c)(ay+b)=ad+bc(ax+c)(ay+b)=ad+bc
  24. 24.Square of a sum (n terms)
    (x1+x2+⋯+xn)2=x12+⋯+xn2+2x1x2+⋯+2xn−1xn(x_1+x_2+\cdots+x_n)^2=x_1^2+\cdots+x_n^2+2x_1x_2+\cdots+2x_{n-1}x_n
  25. 25.Arithmetic Sequence and Series
    an=a1+(n−1)da_n=a_1+(n-1)d
    Sn=a1+a2+⋯+an=n2(a1+an)S_n=a_1+a_2+\cdots+a_n=\frac{n}{2}(a_1+a_n)
  26. 26.Expansion (or factorization) of three binomials
    abc−ab−ac−bc+a+b+c−1=(a−1)(b−1)(c−1)abc-ab-ac-bc+a+b+c-1=(a-1)(b-1)(c-1)
  27. 27.Expansion (or factorization) of two binomials
    a2+ab+ac+bc=(a+b)(a+c)a^2+ab+ac+bc=(a+b)(a+c)
  28. 28.Max-QM-AM-GM-HM-Min Inequality
    x1,…,xn∈R+x_1,\dots,x_n\in\mathbb{R}^+
    max⁡{xi}≥x12+x22+⋯+xn2n≥x1+x2+⋯+xnn≥x1x2⋯xnn≥n1x1+1x2+⋯+1xn≥min⁡{xi}\max\{x_i\}\ge\sqrt{\frac{x_1^2+x_2^2+\cdots+x_n^2}{n}}\ge\frac{x_1+x_2+\cdots+x_n}{n}\ge\sqrt[n]{x_1x_2\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}\ge\min\{x_i\}
    Equality holds if and only if all xi are equal.\text{Equality holds if and only if all }x_i\text{ are equal.}
  29. 29.Integer and Fractional Parts
    [x]: Integer part{x}: Fractional part[x]:\text{ Integer part}\qquad\{x\}:\text{ Fractional part}
    We can write x as:\text{We can write }x\text{ as:}
    x=k+r,k∈Z,  r∈R,  0≤r<1x=k+r,\qquad k\in\mathbb{Z},\;r\in\mathbb{R},\;0\le r<1
  30. 30.Vieta's Formula
    For a quadratic: \text{For a quadratic: }
    ax2+bx+c=0(roots: r1,r2)ax^2+bx+c=0\quad(\text{roots: }r_1,r_2)
    r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1r_2=\frac{c}{a}
    For a cubic: ax3+bx2+cx+d=0(roots: r1,r2,r3)\text{For a cubic: }ax^3+bx^2+cx+d=0\quad(\text{roots: }r_1,r_2,r_3)
    r1+r2+r3=−ba,r1r2+r1r3+r2r3=ca,r1r2r3=−dar_1+r_2+r_3=-\frac{b}{a},\quad r_1r_2+r_1r_3+r_2r_3=\frac{c}{a},\quad r_1r_2r_3=-\frac{d}{a}
    For a general polynomial: anxn+an−1xn−1+⋯+a1x+a0=0(roots: r1,…,rn)\text{For a general polynomial: }a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0\quad(\text{roots: }r_1,\dots,r_n)
    r1+r2+⋯+rn=−an−1anr_1+r_2+\cdots+r_n=-\frac{a_{n-1}}{a_n}
    ∑1≤i<j≤nrirj=an−2an\sum_{1\le i<j\le n}r_ir_j=\frac{a_{n-2}}{a_n}
    ∑1≤i<j<k≤nrirjrk=−an−3an\sum_{1\le i<j<k\le n}r_ir_jr_k=-\frac{a_{n-3}}{a_n}
    r1r2⋯rn−1=(−1)n−1a1anr_1r_2\cdots r_{n-1}=(-1)^{n-1}\frac{a_1}{a_n}
    r1r2⋯rn=(−1)na0anr_1r_2\cdots r_n=(-1)^n\frac{a_0}{a_n}
  31. 31.Signs and Roots of Quadratics
    f(x)=ax2+bx+c,Δ=b2−4acf(x)=ax^2+bx+c,\qquad \Delta=b^2-4ac
    ConditionNumber of Real RootsSign of f(x)f(x)
    Δ>0\Delta>02
    sign(a)0−sign(a)0sign(a)r₁r₂
    Δ=0\Delta=01
    sign(a)0sign(a)r
    Δ<0\Delta<00
    sign(a)
  32. 32.Remainder of Polynomial Division
    A(x)÷B(x):  A(x)=B(x)Q(x)+R(x),deg⁡(R(x))<deg⁡(B(x))A(x)\div B(x):\;A(x)=B(x)Q(x)+R(x),\quad \deg(R(x))<\deg(B(x))
    ⇒ If B(x)=0, then A(x)=R(x)\Rightarrow\text{ If }B(x)=0,\text{ then }A(x)=R(x)
  33. 33.Number of Positive Divisors
    n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
    Number of positive divisors of n:  (α1+1)(α2+1)⋯(αk+1)\text{Number of positive divisors of }n:\;(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_k+1)
  34. 34.Sum of Positive Divisors
    n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
    (1+p1+p12+⋯+p1α1)(1+p2+p22+⋯+p2α2)⋯(1+pk+pk2+⋯+pkαk)(1+p_1+p_1^2+\cdots+p_1^{\alpha_1})(1+p_2+p_2^2+\cdots+p_2^{\alpha_2})\cdots(1+p_k+p_k^2+\cdots+p_k^{\alpha_k})
    =p1α1+1−1p1−1×p2α2+1−1p2−1×⋯×pkαk+1−1pk−1=\frac{p_1^{\alpha_1+1}-1}{p_1-1}\times\frac{p_2^{\alpha_2+1}-1}{p_2-1}\times\cdots\times\frac{p_k^{\alpha_k+1}-1}{p_k-1}
  35. 35.Product of Positive Divisors
    n=p1α1p2α2⋯pkαkn=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
    t=(α1+1)(α2+1)⋯(αk+1)t=(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_k+1)
    Product of positive divisors of n:  nt/2\text{Product of positive divisors of }n:\;n^{t/2}
  36. 36.General Solutions of Trigonometric Equations
    sin⁡α=sin⁡β  ⇒  α=β+360k,  180−β+360k,  k∈Z\sin\alpha=\sin\beta\;\Rightarrow\;\alpha=\beta+360k,\;180-\beta+360k,\;k\in\mathbb{Z}
    cos⁡α=cos⁡β  ⇒  α=±β+360k,  k∈Z\cos\alpha=\cos\beta\;\Rightarrow\;\alpha=\pm\beta+360k,\;k\in\mathbb{Z}
    tan⁡α=tan⁡β  ⇒  α=β+180k,  k∈Z\tan\alpha=\tan\beta\;\Rightarrow\;\alpha=\beta+180k,\;k\in\mathbb{Z}
    cot⁡α=cot⁡β  ⇒  α=β+180k,  k∈Z\cot\alpha=\cot\beta\;\Rightarrow\;\alpha=\beta+180k,\;k\in\mathbb{Z}
  37. 37.Trigonometric Transformations
    sin⁡(90∘−α)=cos⁡αsin⁡(90∘+α)=cos⁡α\sin(90^\circ-\alpha)=\cos\alpha\qquad \sin(90^\circ+\alpha)=\cos\alpha
    cos⁡(90∘−α)=sin⁡αcos⁡(90∘+α)=−sin⁡α\cos(90^\circ-\alpha)=\sin\alpha\qquad \cos(90^\circ+\alpha)=-\sin\alpha
    tan⁡(90∘−α)=cot⁡αtan⁡(90∘+α)=−cot⁡α\tan(90^\circ-\alpha)=\cot\alpha\qquad \tan(90^\circ+\alpha)=-\cot\alpha
    cot⁡(90∘−α)=tan⁡αcot⁡(90∘+α)=−tan⁡α\cot(90^\circ-\alpha)=\tan\alpha\qquad \cot(90^\circ+\alpha)=-\tan\alpha
    sin⁡(180∘−α)=sin⁡αsin⁡(180∘+α)=−sin⁡α\sin(180^\circ-\alpha)=\sin\alpha\qquad \sin(180^\circ+\alpha)=-\sin\alpha
    cos⁡(180∘−α)=−cos⁡αcos⁡(180∘+α)=−cos⁡α\cos(180^\circ-\alpha)=-\cos\alpha\qquad \cos(180^\circ+\alpha)=-\cos\alpha
    tan⁡(180∘−α)=−tan⁡αtan⁡(180∘+α)=tan⁡α\tan(180^\circ-\alpha)=-\tan\alpha\qquad \tan(180^\circ+\alpha)=\tan\alpha
    cot⁡(180∘−α)=−cot⁡αcot⁡(180∘+α)=cot⁡α\cot(180^\circ-\alpha)=-\cot\alpha\qquad \cot(180^\circ+\alpha)=\cot\alpha
    sin⁡(−α)=−sin⁡α\sin(-\alpha)=-\sin\alpha
    cos⁡(−α)=cos⁡α\cos(-\alpha)=\cos\alpha
    tan⁡(−α)=−tan⁡α\tan(-\alpha)=-\tan\alpha
    cot⁡(−α)=−cot⁡α\cot(-\alpha)=-\cot\alpha
  38. 38.Trigonometric Identities
    sin⁡2α+cos⁡2α=1\sin^2\alpha+\cos^2\alpha=1
    sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
    cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
    sin⁡2α=2sin⁡αcos⁡α\sin 2\alpha=2\sin\alpha\cos\alpha
    cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\cos 2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha
    cos⁡3α=4cos⁡3α−3cos⁡α\cos 3\alpha=4\cos^3\alpha-3\cos\alpha
    tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
    cot⁡(α+β)=cot⁡αcot⁡β−1cot⁡α+cot⁡β\cot(\alpha+\beta)=\frac{\cot\alpha\cot\beta-1}{\cot\alpha+\cot\beta}
    sin⁡x+cos⁡x=2sin⁡(x+45∘)\sin x+\cos x=\sqrt{2}\sin(x+45^\circ)
    sin⁡x−cos⁡x=2sin⁡(x−45∘)\sin x-\cos x=\sqrt{2}\sin(x-45^\circ)
  39. 39.Trigonometric Sum-to-Product Identities
    sin⁡α+sin⁡β=2sin⁡(α+β2)cos⁡(α−β2)\sin\alpha+\sin\beta=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
    sin⁡α−sin⁡β=2cos⁡(α+β2)sin⁡(α−β2)\sin\alpha-\sin\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)
    cos⁡α+cos⁡β=2cos⁡(α+β2)cos⁡(α−β2)\cos\alpha+\cos\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
    cos⁡α−cos⁡β=−2sin⁡(α+β2)sin⁡(α−β2)\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)
  40. 40.Trigonometric Product-to-Sum Identities
    sin⁡pcos⁡q=12(sin⁡(p+q)+sin⁡(p−q))\sin p\cos q=\frac{1}{2}(\sin(p+q)+\sin(p-q))
    cos⁡pcos⁡q=12(cos⁡(p+q)+cos⁡(p−q))\cos p\cos q=\frac{1}{2}(\cos(p+q)+\cos(p-q))
    sin⁡psin⁡q=12(cos⁡(p−q)−cos⁡(p+q))\sin p\sin q=\frac{1}{2}(\cos(p-q)-\cos(p+q))