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
` T_n = a + \ left( n - 1 \ right) d `
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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` 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 } `
