Sequences, Progressions, Series, Means and Averages

Averages

  • SimpleAverage=SumofElementsNumberofElemnts

  • WeightedAverage=w1x1+w2x2+w3w3++wnwnw1+w2+w3++wn

Mean

  • ArithmeticMean=x1+x2+x3++xnn

  • GeometricMean=x1×x2×x3××xnn

  • HarmonicMean=n(1x1+1x2+1x3++1xn)

For two numbers a and b, HarmonicMean=2aba+b

Arithmetic Progression

  • An Arithmetic Progression (A. P.) with first term ‘a’ and Common Difference ‘d’ is given by: [a],[(a+d)],[(a+2d)],.........,[a+(n1)d] and nth term, Tn=a+(n1)d

  • Tn=a+(n1)d

  • Sn=n2[2a+(n1)d]

  • Sum of first ‘n’ terms, Sn =n/2 (First Term + Last Term)

Geometric Progression

  • A Geometric Progression (G. P.) with first term ‘a’ and Common Ratio ‘r’ is given by: a,ar, ar2 , ar3 ,,ar n-1 where n th term, Tn=arn1

  • Tn=arn1

  • Sn=(arn1)(r1)

  • S=a(r1),forr<1

Harmonic Progression

  • Tn=1a+(n1)d

Sum of Important Series

  • Sum of first n natural numbers: 1+2+3+4++n=n(n+1)2

  • Sum of the squares of the first n natural numbers: 12+22+32+42++n2=n(n+1)(2n+1)6

  • Sum of the cubes of the first n natural numbers: (13+23+33+.........+n3)=[n2(n+1)2]4