Introduction
The ordered collection of objects is called sequence. The sequence having specified patterns is called progression. In this chapter besides discussing about the arithmetic progression, we will also discuss about the geometric progression and arithmetic-geometric progression. The various numbers occurring in the sequence is called term of the sequence. A sequence having finite number of terms is called finite sequence, where as the sequence having infinite number of terms is called infinite sequence.
The real sequence is that sequence whose range is a subset of the real number.
A series is defined as the expression denoting the sum of the terms of the sequence. The sum is obtained after adding the terms of the sequence. If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},---,{{a}_{n}}\] is a sequence having n terms, then the sum of the series is given by,
\[\sum\limits_{n=1}^{m}{{{a}_{n}}={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+----{{a}_{n}}}\]
Arithmetic Progression (A.P.)
A sequence is said to be in A.P, if the difference between the consecutive terms is a constant. The difference between the consecutive terms of an AP is called common difference and any general term is called nth term of the sequence.
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},---,{{a}_{n}}\] be the nth terms of the sequence in A.P., then nth terms of the sequence is given by \[{{a}_{n}}=a+\left( n-1 \right)d\],
Where 'a' is the first term of the sequence
'd' is the common difference
'n' is the number of terms of the sequence.
Sum of N-Terms of the A.P.
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},---,{{a}_{n}}\] be the n terms of the sequence in A.P., then the sum of n-terms of the sequence is given \[{{S}_{n}}=\frac{n}{2}\left[ 2a+(n-1)d \right]\]
Arithmetic Mean
If 'a' and 'b' are any two terms of A.P., then the arithmetic mean more...
Geometric Progression (G.P.)
A sequence is said to be in G.P. if the ratio between the consecutive terms is constant. The sequence \[{{a}_{1}},{{a}_{2}},{{a}_{3}},---,{{a}_{n}}\] is said to be in G.P. if the ratio of the consecutive term is a constant.
If 'r' is the common ratio, then the nth term of the sequence is given by \[{{a}_{n}}=a\,\,{{r}^{n-1}}\]
The sum of n terms of the G.P. sequence is given by
\[{{S}_{n}}=\frac{a({{r}^{n}}-1)}{r-1}If\,\,r\,>\,1\,and\,{{S}_{n}}=\frac{a(1-{{r}^{n}})}{1-r}if\,r\,>1\]
Sum to infinity is a G.P. series is given by \[{{S}_{\propto }}=\frac{a}{1-r}.\]
Geometric Mean (G.M.)
If 'a' and 'b' are any two terms of G.P., then the geometric mean is given by \[GM=\sqrt{ab}\]
Properties of GP
(a) If each term of GP is multiplied or divided by a constant, then the resulting sequence is also in GP.
(b) If each term of the GP is raised to the same power more...
Harmonic Progression (H.P.)
The sequence is said to be in H.P. If the reciprocal of its terms gives the A.P. It has got wide application in the field of geometry and theory of sound. The questions are generally solved by inverting the terms and using the property of arithmetic progression.
Three numbers a, b, c are said to be in HP if,
\[\frac{a}{c}=\frac{a-b}{a-c}\]
Harmonic Mean (HM)
If 'a' and 'b' be any two terms, then their harmonic mean is given by \[HM=\frac{2ab}{a+b}\].
Relation between AM, GM, and HM
Since we know that,
\[AM=\frac{a+b}{2},\,\,GM=\sqrt{ab}\,and\,HM=\frac{2ab}{a+b}\]
Then,
\[AM\times HM=\frac{a+b}{2}\times \frac{2ab}{a\times b}=ab={{G}^{2}}\]
\[AM\times HM=G{{M}^{2}}\]
Form the above relation we can say that AM > GM and GM is intermediate value between AM and HM, therefore GM > HM. Hence we can say that AM > GM > HM.
Also the relation between A and more...
