Common non-formulae:
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- Sum of kth(k≠pn,p is an integer) power of nth roots of unity = 0
- Product of nth roots of unity = 1 if n is odd , −1 is n is even
- nth term of an AP = a + (n−1)d
- nth term of a GP = arn−1
- nth term of a HP = 1/(a+(n−1)d)
- nth term of a AGP = [a+(n−1)d]r(n−1)
- HM ≤ GM ≤ AGM ≤ AM ≤ QM {QM = √(Σx2)/n}
- Sum of first n natural numbers = n(n+1)/2
- Sum of squares of the first n natural numbers = n(n+1)(2n+1)/6
- 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)
- Weighted arithmetic mean, AW = (w1a1 + w2a2 + w3a3 + ... + wnan)/(w1 + w2 + w3 + ... + wn)
- Weighted geometric mean, GW = (a1w1 * a2w2 * a3w3 * ... * anwn)[1/(w1 + w2 + w3 + ... + wn)]
- Weighted harmonic mean, HW = (w1 + w2 + w3 + ... + wn)/(w1/a1 + w2/a2 + w3/a3 + ... + wn/an)
- 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
Formulae:
- Number of power sets = 2n
- A−B = B'−A'
- A⊆B = B'⊆A'
- (A∪B)' = A'∩B'
- (A∩B)' = A'∪B'
- A−(B∪C) = (A−B)∩(A−C)
- n(A∪B) = n(A) + n(B) − n(A∩B)
- 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|
- Insertion of n AMs , d = (b−a)/n+1 , Ak = a + kd
- Insertion of n GMs , r = (b/a)(1/(n+1)) , Gk = ark
- Insertion of n HMs , D = (a−b)/[(n+1)ab] , 1/Hk = (1/a) + kD
- Sum of n terms in AP , S = (n/2)[2a+(n−1)d] = (n/2)[a+an]
- Sum of n terms in GP , S = a(1 − rn)/(1−r)
- Sum of infinite terms in GP {when |r|<1}, S∞ = a/(1−r)
- 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
- nPr = <n,r> = n!/(n−r)!
- nCr = [n,r] = n!/[r!(n−r)!]
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