Below are the list of all sequence and series formulas;
ARITHMETIC SEQUENCE OR
ARITHMETIC PROGRESSION (A.P):
To find the nth term of Arithmetic Progressions are
Tn=a+
Where;
a = first term ,
d = difference ,
n = number of terms ,
T_n = nth term or last term or general term or rule of formation
To find the number of terms of Arithmetic Progressions are
n = \ frac{ T_n - a } d+1
To find the difference of Arithmetic Progressions are
Difference = Second term - First term
OR
d = \ frac{ T_n - a } { n-1 }
ARITHMETIC SERIES:
To find the sum of Arithmetic Progressions are
S_n = \ frac{ n } { 2 } \ [ 2 a + (n - 1 ) d ]
OR
S_n = \ frac{ n } { 2 } \ ( a + l )
ARITHMETIC MEANS:
The formula to find the arithmetic mean are given as;
A.M = \ frac{ a+b } { 2 }
When you have more than one arithmetic mean, first you can find d by using this formula;
d = \ frac{ b-a } { n+1 }
Then, you can find n(number of) arithmetic means;
A_1 = a + 1 d
A_2 = a + 2 d
A_3 = a + 3 d
.
.
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A_n = a + n d
WHEN TERMS ARE IN ARITHMETIC
PROGRESSION:
When three numbers are in Arithmetic Progression (A.P)
First number are in Arithmetic Progression (A.P) = a - d
Second number are in Arithmetic Progression (A.P) = a
Third number are in Arithmetic Progression (A.P) = a + d
When four numbers are in Arithmetic Progression (A.P)
First number are in Arithmetic Progression (A.P) = a - 3 d
Second number are in Arithmetic Progression (A.P) = a - d
Third number are in Arithmetic Progression (A.P) = a + d
Fourth number are in Arithmetic Progression (A.P) = a + 3 d
When five numbers are in Arithmetic Progression (A.P)
First number are in Arithmetic Progression (A.P) = a - 2 d
Second number are in Arithmetic Progression (A.P) = a - d
Third number are in Arithmetic Progression (A.P) = a
Fourth number are in Arithmetic Progression (A.P) = a + d
Fifth number are in Arithmetic Progression (A.P) = a + 2 d
GEOMETRIC SEQUENCE OR
GEOMETRIC PROGRESSION (G.P):
To find the nth term of Geometric Progressions are
T_n = a r^{ n-1 }
Where;
a = first term ,
r = ratio ,
n = number of terms ,
T_n = nth term or last term or general term or rule of formation
To find the ratio of Geometric Progressions are
ratio = Second term \div First term
GEOMETRIC SERIES:
There are two formulas to find the sum of geometric series formulas;
When r > 1
S_n = \ frac{ a \ left( r ^ n - 1 \ right) } { r - 1 }
OR
S_n = \ frac{ r l - a } { r - 1 }
When r < 1
S_n = \ frac{ a \ left( 1 - r ^ n \ right) } { 1 - r }
OR
S_n = \ frac{ a - r l } { 1 - r }
INFINITE GEOMETRIC SERIES:
The formula to find the sum of infinite geometric series are given as;
S = \ frac{ a } { 1-r }
GEOMETRIC MEANS:
The formula to find the geometric mean are given as;
G.M = \ pm \ sqrt { a b }
When you have more than one geometric mean, first you can find r by using this formula;
r = \ left( \ frac b a \ right) ^ \frac { 1 } { n+1 }
Then, you can find n(number of) geometric means;
G_1 = a r ^ 1
G_2 = a r ^ 2
G_3 = a r ^ 3
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.
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G_n = a r ^ n
WHEN TERMS ARE IN GEOMETRIC
PROGRESSION:
When three numbers are in Geometric Progression (G.P)
First number are in Geometric Progression (G.P) = \frac a r
Second number are in Geometric Progression (G.P) = a
Third number are in Geometric Progression (G.P) = a r
When four numbers are in Geometric Progression (G.P)
First number are in Geometric Progression (G.P) = \ frac a { r^3 }
Second number are in Geometric Progression (G.P) = \frac a r
Third number are in Geometric Progression (G.P) = a r
Fourth number are in Geometric Progression (G.P) = a r^3
When five numbers are in Geometric Progression (G.P)
First number are in Geometric Progression (G.P) = \ frac a { r^2 }
Second number are in Geometric Progression (G.P) = \frac a r
Third number are in Geometric Progression (G.P) = a
Fourth number are in Geometric Progression (G.P) = a r
Fifth number are in Geometric Progression (G.P) = a r^2
HARMONIC SEQUENCE OR
HARMONIC PROGRESSION (H.P):
To find the nth term of Harmonic Progressions are
T_n = \ frac{ a b } { b + \ left( n - 1 \ right) \ left( a - b \ right) }
HARMONIC MEANS:
The formula to find the harmonic mean are given as;
H.M = \ frac{ 2 a b } { a+b }
When you have more than one harmonic mean, first you can find d by using this formula;
d = \ frac{ ( a - b ) } { left( n - 1 \ right) \ a b }
Then, you can find n(number of) harmonic means;
H_1 = \ frac{ \ left( n + 1 \ right) ab } { a + nb }
H_2 = \ frac{ \ left( n + 1 \ right) ab } { 2a + \ left( n - 1 \ right) b }
H_3 = \ frac{ \ left( n + 1 \ right) ab } { 3a + \ left( n - 2 \ right) b }
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H_n = \ frac{ \ left( n + 1 \ right) ab } { na + b }
