Sequences and Series

Arithmetic Progression
tn= a + (n-1)d

Sn = n/2 [2a + (n-1)d] =n/2 [a + tn]

tn = Sn - Sn-1

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a2 - a1 = an - an-1 = constant

a1 + an= a2 + an-1 = constant

a1 + k, a2 + k , a3 + k ,......., an + k are also in A.P.

a1k, a2k , a3k ,......., ank are also in A.P.

a1, a3, a5 are also in AP

a2, a4, a6 are also in AP

{terms falling after equal intervals are in AP}

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3 consecutive terms :

4 consecutive terms :

5 consecutive terms :

AM = (a + b) / 2  ; for two numbers

AM = (a1 + a2 + a3 + ..... + an) / n (for n numbers)

Geometric Progression
tn = arn-1

Sn= a(rn-1) / (r - 1) if r > 1

S∞= a / (1 - r) if 1 < r

tn = Sn /Sn-1

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a2/a1 = a3/a2 = an/an-1= common ratio

a1an = a2an-1 = constant

a1 + k, a2 + k , a3 + k ,......., an + k are also in A.P.

a1k, a2k , a3k ,......., ank are also in G.P.

a1, a3, a5 are also in GP

a2, a4, a6 are also in GP

{terms falling after equal intervals are in GP}

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3 consecutive terms :

4 consecutive terms :

5 consecutive terms :

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GM = √(ab)     for two numbers

GM = √(a1aca3....an)n   (for n numbers)

Harmonic Progression
tn = 1 / A.P.

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(1 / an) - (1 / an-1)undefined = constant

HM = 2ab / (a+b)

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3 consecutive terms :

4 consecutive terms :

5 consecutive terms :

Arithmetic Mean
AM = (a + b)/2

Geometric Mean
GM = √(ab)

Harmonic Mean
HM = 2ab / (a+b)

Relation between AM,GM,HM
AH = G2

Aritmetico - Geometrico Series
a.b + (a+d)br + (a+2d)br2........ (a+(n-1)d)brn-1

is called an Arithmetico Geometrico Series

For such series, we multiply the equation with r and then subtract it from the original equation.

Sum of Finite Series
Σn0 = n

Σn = n(n-1) / 2

Σn2 = n(n-1)(2n-1) / 6

Σn3 = n(n-1)undefined/ 2]2

Note : For other summations, convert the expression into the above given expressions.

Infinite Series
Sum = a / (1- r)

Special Series
Special Series works on the Principle of Binomial Expansion.

Algebraic Expansions

(1-x)-1 = 1 + x + x2 + x3 + ...... ∞

(1+x)-1 = 1 - x + x2 - x3 + ..... ∞

(1 - x)-2 = 1+ 2x + 3x2 + 4x3 + .....∞

(1+ x)-2 = 1 - 2x + 3x2 - 4x3 + ....∞

Exponential Series

ex = 1 + x/1! + x2/2! + x3/3!......∞

e-x = 1 - x/1! + x2/2! - x3/3!......∞

Logarithmic Series

log(1+x) = x - x2/2undefined+ x3/3 - ....... ∞

log (1-x) = - x - x2/2undefined- x3/3 - ....... ∞