Common non-formulae:

• $\left|\left|x\right|-\left|y\right|\right|\le \left|x\pm y\right|\le \left|x\right|+\left|y\right|$
• $\text{Sum of k^th(k≠pn,p is an integer) power of n^th roots of unity = 0}$
• Sum of k^th(k≠pn,p is an integer) power of n^th roots of unity = 0
• Product of n^th roots of unity = 1 if n is odd , −1 is n is even
• 1. n^th term of an AP = a + (n−1)d
  2. n^th term of a GP = arⁿ⁻¹
  3. n^th term of a HP = 1/(a+(n−1)d)
  4. n^th term of a AGP = [a+(n−1)d]r⁽ⁿ⁻¹⁾
• HM ≤ GM ≤ AGM ≤ AM ≤ QM {QM = √(Σx²)/n}
• 1. Sum of first n natural numbers = n(n+1)/2
  2. Sum of squares of the first n natural numbers = n(n+1)(2n+1)/6
  3. Sum of cubes of the first n natural numbers = [n(n+1)/2]²
• Sum of first n powers of a = (a−aⁿ⁺¹)/(1−a)
• 1. Weighted arithmetic mean, A_W = (w₁a₁ + w₂a₂ + w₃a₃ + ... + w_na_n)/(w₁ + w₂ + w₃ + ... + w_n)
  2. Weighted geometric mean, G_W = (a₁^w₁ * a₂^w₂ * a₃^w₃ * ... * a_n^w_n)^{[1/(w₁ + w₂ + w₃ + ... + w_n)]}
  3. Weighted harmonic mean, H_W = (w₁ + w₂ + w₃ + ... + w_n)/(w₁/a₁ + w₂/a₂ + w₃/a₃ + ... + w_n/a_n)
• Number of ways to complete a and b = number of ways to complete a * number of ways to complete b
• Number of ways to complete either a xorOne or the other but not both b = number of ways to complete a + number of ways to complete b
• n! = n * (n−1) * (n−2) * ... * 3 * 2 * 1
• Number of permutation with identical elements = number of permutation with all distinct elements/[∏[(number of duplicates of ith type)!]]
• [n,r] + [n,r−1] = [n+1,r]
• [n,r] = [n,n−r]
• ∑[n,r] = 2ⁿ
• ∑_r>0([n,r]) = 2ⁿ − 1

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Formulae:

• Number of power sets = 2ⁿ
• 1. A−B = B'−A'
  2. A⊆B = B'⊆A'
  3. (A∪B)' = A'∩B'
  4. (A∩B)' = A'∪B'
  5. A−(B∪C) = (A−B)∩(A−C)
  6. n(A∪B) = n(A) + n(B) − n(A∩B)
  7. n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
• √a+bi = ±[{sqrt(a² + b²)+a}/2] ± sgn(b)*isqrt[{sqrt(a² + b²)−a}/2]
• |z₁ ± z₂| = |z₁|² + |z₂|² ±Re(z₁z̅₂)
• Distance = |z₁−z₂|
• 1. Insertion of n AMs , d = (b−a)/n+1 , A_k = a + kd
  2. Insertion of n GMs , r = (b/a)^(1/(n+1)) , G_k = ar^k
  3. Insertion of n HMs , D = (a−b)/[(n+1)ab] , 1/H_k = (1/a) + kD
• 1. Sum of n terms in AP , S = (n/2)[2a+(n−1)d] = (n/2)[a+a_n]
  2. Sum of n terms in GP , S = a(1 − rⁿ)/(1−r)
  3. Sum of infinite terms in GP {when |r|<1}, S_∞ = a/(1−r)
  4. Sum of n terms of in AGP , S(1−r) = a + dr(1 − rⁿ⁻¹)/(1−r) − [a+(n−1)d]rⁿ
• ∑_{1≤i≤j≤n}[ƒ(i)ƒ(j)] = [∑∑(ƒ(i)ƒ(j))+∑_i=j(ƒ(i)ƒ(j))]/2
• ⁿP_r = <n,r> = n!/(n−r)!
• ⁿC_r = [n,r] = n!/[r!(n−r)!]

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Tips:

• Set theory is useful in number category questions
• Remember flipping inequalities
• Domain of a function can help to eliminate extraneous roots
• If a root is common between 2 quadratics then eliminate the x² term to find it
• For cubic, guess a root and divide by it for solving
• Expressing a series as a telescopic + another part can be helpful for calculating sum of its terms
• If inequalities using sum and product are given then consider the AM,GM inequality
• It is often simpler to use the procedure of multiplying the sum of an AGP by r and subtracting from it, i.e., S*r − S, to find the sum of an AGP
• For number of permutations keeping 2 types of objects separated, it is best to do it manually and carefully utilizing the gaps where the objects can be placed
• For number of combinations of objects of a group together, break it into multiple cases
• For number of permutations it is best to find number of combinations and then arrange to avoid errors
• For circular permutations, there is no difference in where where the first object is placedso there are n times less permutations
• If there is a difference between the positions in circular permutation then it must be treated as a linear permutation