AP GP and HP

An arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
For example, the sequence 3, 5, 7, 9, 11,... is an arithmetic progression with common difference 2.

Arithmetic progression property:

a₁ + a_n = a₂ + a_n-1 = ... = a_k+a_n-k+1

Formulae for the n-th term can be defined as:

a_n = ¹/₂(a_n-1 + a_n+1)

If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the n-th term of the sequence is given by

a_n = a₁ + (n - 1)d, n = 1, 2, ...

The sum S of the first n values of a finite sequence is given by the formula:

S = ¹/₂(a₁ + a_n)n, where a₁ is the first term and a_n the last.

or

S = ¹/₂(2a₁ + d(n-1))n

 

GP

 

S = a + a.r + a.r² + a.r³ + ... + a.r^(n-1) [Equation 1]

Now multiply each side of the equation by r:

S.r = a.r + a.r² + a.r³ +... + a.r^(n) [Equation 2]

Now subtract Equation 1 from Equation 2: [Note that the terms from a.r to a.r^(n-1) are in both series and cancel out]

S.r - S = a.r^n - a

S.(r-1) = a.r^n - a

= a(r^n - 1)

So,

S = a.(r^n - 1) ÷ (r - 1)

In cases like Example 2, where r < 1, it is often written as:

S = a.(1 - r^n) ÷ (1 - r)

Convergence of GP

If |r| < 1 then a_n -> 0, when n -> ∞ So the sum S of such a infinite geometric progression is:

S =  1 / (1-r)

which is valid only for |r| < 1

 

Formulae

ü The sum of first n natural numbers = n(n+1)/2

ü The sum of squares of first n natural numbers is n(n+1)(2n+1)/6

ü The sum of cubes of first n natural numbers is (n(n+1)/2)²/4

ü The sum of first n even numbers= n (n+1)

ü The sum of first n odd numbers= n²