Progression

Sequence:
A sequence represents numbers formed in succession and arranged in a fixed order defined by a certain rule.


Airthmetic Progression ( A.P.):
It is a type of sequence where each number/term(except first term) differs from its preceding number by a constant. This constant is termed as common difference.


A.P. Terminologies:
• First number is denoted as 'a'.
• Common difference is denoted as 'd'.
• nth number is denoted as 'Tn'.
• Sum of n number is denoted as 'Sn'.


A.P. Examples:
• 1, 3, 5, 7, ... is an A.P. where a = 1 and d = 3 - 1 = 2.
• 7, 5, 3, 1, - 1 ... is an A.P. where a = 7 and d = 5 - 7 = -2.


General term of A.P:
Tn = a + (n - 1)d
Where a is first term, n is count of terms and d is the difference between two terms.


Sum of n terms of A.P:
Sn = n/2 [2a + (n - 1)d]
Where a is first term, n is count of terms and d is the 2 difference between two terms. There is another
Sn = n/2(a + l)
Where a is first term, n is count of terms, l is the last term.


Geometrical Progression(G.P.):
It is a type of sequence where each number/term(except first term) bears a constant ratio from its preceding number. This constant is termed as common ratio.


G.P. Terminologies:
• First number is denoted as 'a'.
• Common ratio is denoted as 'r'.
• nth number is denoted as 'Tn'.
• Sum of n number is denoted as 'Sn'.


G.P. Examples:

• 3, 9, 27, 81, ... is a G.P. where a = 3 and r = 9/3= 3.

• 81, 27, 9, 3, 1 ... is a G.P. where a = 81 and r = 27/81 = 1/3


General term of G.P:
Tn = ar(n-1)
Where a is first term, n is count of terms, r is the common ratio

Sum of n terms of G.P.: Sn = a(1 - rn)/(1 - r)
Where a is first term, n is count of terms, r is the common ratio and r < 1.

There is another variation of the same formula:
Sn = a(rn - 1)/(r - 1)
Where a is first term, n is count of terms, r is the common ratio and r > 1.


Arithmetic Mean:
Arithmetic mean of two numbers a and b is (a+b)/2


Geometric Mean:
Geometric mean of two numbers a and b is ?ab


General Formulaes:

1 + 2 + 3 + ....... + n = n/2 (n+1)

12 + 22 + 32 + ... + n2 = n/6(n+1)(2n+1)

13 + 23 + 3%3 + ... + n%3 = [n/2 (n+1)]2