ARITHMETIC AND GEOMETRIC PROGRESSIONS SUMMARY FOR CA FOUNDATION EXAMS.

- Sequence: An ordered collection of numbers a
_{1}, a_{2}, a_{3}, a_{4}, …………….., a_{n}, …………….. is a sequence if according to some definite rule or law, there is a definite value of a_{n}, called the term or element of the sequence, corresponding to any value of the natural number n. - An expression of the form a
_{1}+ a_{2}+ a_{3}+ ….. + a_{n}+ ………………………. which is the sum of the elements of the sequenece { a_{n}} is called a series. If the series contains a finite number of elements, it is called a finite series, otherwise called an infinite series. - Arithmetic Progression: A sequence a
_{1}, a_{2},a_{3}, ……, an is called an Arithmetic Progression (A.P.) when a_{2}– a_{1}= a_{3}– a_{2}= ….. = a_{n}– a_{n}–1. That means A. P. is a sequence in which each term is obtained by adding a constant d to the preceding term. This constant ‘d’ is called the common difference of the A.P. If 3 numbers a, b, c are in A.P., we say

b – a = c – b or a + c = 2b; b is called the arithmetic mean between a and c.

n^{th} term ( t_{n} ) = a + ( n – 1 ) d,

Where a = First Term

D = Common difference= t_{n}– t_{n-1}

Sum of n terms of AP=

- Sum of the first n terms: Sum of 1st n natural or counting numbers

S = n( n + 1 )/2

Sum of 1st n odd number : S = n^{2}

Sum of the Squares of the first, n natural numbers:

sum of the squares of the first, n natural numbers is

- Geometric Progression (G.P). If in a sequence of terms each term is constant multiple of the proceeding term, then the sequence is called a Geometric Progression (G.P). The constant multiplier is called the common ratio

- Sum of first n terms of a G P:

Sn = a ( 1 – r_{n}) / ( 1 – r ) when r < 1

Sn = a ( r_{n}– 1 ) / ( r – 1 ) when r > 1

Sum of infinite geometric series

- A.M. of a & b is = ( a + b ) /2
- If a, b, c are in G.P we get b/a = c/b => b
^{2}= ac, b is called the geometric mean between a and c