Progressions

Arithmetic Progression :

A sequence of numbers is said to be in an Arithmetic progression if the difference between any two consecutive terms is always the same. 2, 4, 6, 8, 10 is an AP series because they have a common difference (4-2=6-4=8-6=10-8)

Few facts about AP

• Initial Term (a): first number in the series
• Common difference (d): The value by which consecutive terms increase or decrease
• The behavior of the AP depends on the common difference d. If d > 0, then the members (terms) will grow towards positive infinity. If d < 0, then the members (terms) will grow towards negative infinity.
• nth term of AP : a + (n-1)*d
• Arithmetic Mean : Sum of all terms in the AP / Number of terms in the AP. AM of 2 terms: (a+b) / 2
• sum of ‘n’ terms : 0.5 n (first term + last term) = 0.5 n [ 2a + (n-1) d ].

Geometric Progression.

A sequence of numbers is said to be in a Geometric progression if the ratio of any two consecutive terms is always same. 2, 4, 8, 16 is a GP because they have common ratio. (4 / 2 = 8 / 4 = 16 / 8 = 2).

Few facts about GP

• Initial Term (a): first number in the series
• Common ratio (r): The ratio between a term in the sequence and the term before it
• The behaviour of a geometric sequence depends on the value of the common ratio.
  • r > 0 -> the terms will all be the same sign as the initial term.
  • r < 0 -> the terms will alternate between positive and negative.
  • r > 1 -> exponential growth towards positive or negative infinity (depending on the sign of the initial term).
  • r = 1 -> progression is a constant sequence
  • -1<=r<=1, (but not zero) -> exponential decay towards zero.
  • r = -1 -> progression is an alternating sequence.
  • r < -1 -> for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.
• nth term of GP : a*r ^ (n-1).
• Geometric Mean (GM) : nth root of product of n terms in the GP. GM of 2 terms : sqrt(a*b)
• Sum of ‘n’ terms of a GP (r < 1) : [a (1 – r^n)] / [1 – r].
• Sum of ‘n’ terms of a GP (r > 1) : [a ((r^n) – 1)] / [r – 1].
• Sum of infinite terms of a GP (r < 1): (a) / (1 – r).

Harmonic Progression

A harmonic progression is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. If a,b,c,d are in an arithmetic progression, so its reciprocals 1/a, 1/b, 1/c, 1/d are in harmonic progression.

Few facts about HP

• Harmonic Mean(HM) of 2 terms: 2ab/(a+b)
• If A,G,H are AM, GM, HM where A>=G>=H : AH= G^2 . A,G,H are in GP.

Mean and Median

Mean

Average of given set of data. Mean = Total sum / n. Its a very popular measure of central tendency, which can be used with both discrete and continous data, most often with continous data.

Median

Middle value of set of data . To get median value, numbers must be in sorted order.

• N is odd : Median = middle number from sorted data :(N+1) / 2th value
• N is even : Median = average of two middle values. : Average of (N/2)th and {(N/2) + 1}th value.
• If the user adds or multiplies a constant to every value, then mean and median will also be added or multiplied by the same constant.

Quadratic Equations

ax^2 + bx + c = 0 Where a,b and c are real known values and, a can’t be zero.

Roots of equation :

Roots are values for which the equation satisfies the given condition. the roots of equation x^2 - 7x - 12 = 0 are 3 and 4 respectively. If we replace the value of x by 3 and 4 individually in the equation, the equation will evaluate to zero.

We can have 2 roots for a QE:

r1,r2 = (-b ± √(b^2 - 4ac))/2a

• Determinant (D) : (b^2 - 4ac)

There are 3 different cases while finding the roots:

• D < 0 0r b^2 < 4ac : roots are imaginary (not real)
  • for equation, x^2 + x + 1 : r1 = -0.5 + i1.73205, r2 = -0.5 - i1.73205
• D = 0 or b^2 = 4ac : roots are real and equal (r1= r2)
  • for eqn, x^2 - 2x + 1 : r1 = 1 and r2 = 1
• D > 0 or b^2 > 4ac : roots are real and different
  • for eqn, x^2 - 7x - 12 : r1 =3 and r2 = 4

Permutations and Combinations

Permutation: Permutation is defined as arrangement of ‘r’ things that can be done out of total ‘n’ things

Denoted by nPr = n!/(n-r)!

Combination: Combination is defined as Selection of ‘r’ things that can be done out of total ‘n’ things

Denoted by nCr = n!/r!(n-r)!

Modular Arithmetic

The remainder obtained after division operation on two operands is known as modulo arithmetic.