Progressions Formulas, Tricks and Shortcuts

Arithmetic Progressions Page - 2
Geometric Progressions Page - 3
Progressions Important Questions Page - 6

Sequences, following specific patterns are called progressions. Arithmetic progression (A.P). In this Chapter, besides discussing more about A.P.; arithmetic mean, geometric mean, relationship between A.M. and G.M., special series in forms of sum to n terms of consecutive natural numbers, sum to n terms of squares of natural numbers and sum to n terms of cubes of natural numbers will also be studied.

Sequences

Let the number of person’s ancestors for the first, second, third, ..., tenth generations are 2, 4, 8, 16, 32, ..., 1024. These numbers form what we call a sequence. Consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division. In this process we get 3,3.3,3.33,3.333, ... and so on. These quotients also form a sequence. The various numbers occurring in a sequence are called its term . We denote the terms of a sequence by a_{1},a_{2},a_{3},.....,a_{n} etc, the subscripts denote the position of the term. The n^{th} term is the number at the position of the sequence and is denoted by a_{n} The n^{th} term is also called the general term of the sequence.

A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends .Often, it is possible to express the rule, which yields the various terms of a sequencein terms of algebraic formula. Consider for instance, the sequence of even natural number 2, 4, 6,...
Here, a_{1}=2=2\times 1,a_{2}=4=2\times 2,a_{3}=6=2\times 3,........a_{23}=46=2\times 23,a_{24}=48=2\times 24 , and so on. In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern, but the sequence is generated by the recurrence relation given by.
a_{1}=a_{2}=1,a_{3}=a_{1}+a_{2},a_{n}=a_{n+2}+a_{n-1},n> 2
Above sequence is called Fibonacci sequence

Series

a_{1},a_{2},a_{3},.....,a_{n} be a given sequence. Then, the expression a_{1}+a_{2}+a_{3}+.....+a_{n} s called the series associated with the given sequence. The series is finite or infinite according as the given sequence is finite or infinite. Series are often represented in compact form called sigma notation as means indicating the summation involved .Thus , the series a_{1}+a_{2}+a_{3}+.....+a_{n} is abbreviated . as \sum_{k-1}^{n}

Note: When the series is used, it refers to the indicated sum not to the sum itself. For example, 1 + 3 + 5 + 7 is a finite series with four terms. When we use the phrase “ sum of a series,” we will mean the number that results from adding the terms, the
sum of the series is 16.

 

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