CA Foundation Arithmetic and Geometric Progressions Summary

Arithmetic and Geometric Progressions Summary for CA Foundation November 2020 Exams

• Sequence: An ordered collection of numbers a₁, a₂, a₃, a₄, …………….., 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 a₁+ a₂+ a₃+ ….. + 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 a₁, a₂ ,a₃, ……, an is called an Arithmetic Progression (A.P.) when a₂ – a₁ = a₃ – a₂ = ….. = 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 = n²
  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 => b²= ac, b is called the geometric mean between a and c