11th Class Mathematics Sequence and Series Sequence and Series (A.P., G.P. and H.P.)

                                             SEQUENCE AND SERIES (A.P., G.P. AND H.P.)

INTRODUCTION

SEQUENCE                                

A systematic umbers according to a given rule is called a sequence: The sum of terms of a sequence is called a series.

The first term of a sequence is denoted as \[{{T}_{1}}\], second term is denoted as \[{{T}_{2}}\], and so on. The nth term, of sequence is denoted by \[{{T}_{n}}\]. It is also referred to as he general term of the sequence.

Finite and Infinite Sequences

1. A sequence containing finite number of terms is called a finite sequence.

Example: 1, 9, 17, 25, 33, is a finite sequence of 5 terms.

2. A sequence consisting of infinite numbers of terms is called an infinite sequence.

Example: 3, 6, 9, 12, 15..................... up to infinite number of terms.

If a sequence is given, then we can find its nth term and if the nth term of a sequence is given we can find the terms of the sequence.

Elementary question 1

Find the nth term of the sequence 4, 7, 10, 13, 16,..............

Answer. From observation, \[{{T}_{1}}=3\times 1+1\]                  \[{{T}_{2}}=3\times 2+1\]                    \[\therefore {{T}_{n}}=3n+1\]

SERIES

The sum of term of a sequence is called the series of the corresponding sequence.

Example: 2 + 4 + 6 +...... + 2n is a finite series of first n even natural number

The sum of first n terms of series is denoted by \[{{S}_{n}}\].

Here, \[{{S}_{n}}={{T}_{1}}+{{T}_{2}}+......+{{T}_{n}}\]

Here, \[{{S}_{1}}={{T}_{1}}\];\[{{S}_{2}}={{T}_{1}}+{{T}_{2}}\];             \[{{S}_{3}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}\]; \[{{S}_{4}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}+{{T}_{4}}\]……………

\[\therefore {{S}_{n}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}+.....+{{T}_{n}}\]

And, we have,    \[{{S}_{2}}-{{S}_{1}}={{T}_{2}}\]

                        \[{{S}_{3}}-{{S}_{2}}={{T}_{3}};\]   \[{{S}_{4}}-{{S}_{2}}={{T}_{4}}\]and so on.

Similarly,

            \[{{S}_{n}}-{{S}_{n-1}}={{T}_{n}}\]

Arithmetic mean (A.M.): If a and c are any two terms of an A.P, then the arithmetic mean (A.M.) ‘b’ is given by, \[b=\frac{a+c}{2}\].

When three numbers a, b, c arranged in order of there is creasing values differ by a fixed number ‘d’, then they are said they are said to be in arithmetic progression  (A, P,) and the fixed number ‘d’ is called common difference.

\[a+d=b\]

\[b+d=c\]

Or         \[bacb\]

Or         \[b=\frac{a+c}{2}\]

If ‘n’ numbers \[{{x}_{1}},{{x}_{2}}......{{x}_{n}}\] are in A. P. then

\[{{x}_{2}}=\text{ }{{\text{x}}_{1}}+d\]

\[{{x}_{3}}=\text{ }{{x}_{1}}+2d\]

\[{{x}_{n}}=\text{ }{{x}_{1}}(n-1)d\]

\[\therefore \]\[{{n}_{th}}\] term of A. P. \['{{X}_{n}}'\] = first term \[+\left( n-1 \right)\times \]common difference

Sum of an A. P. up to ‘n’ terms

\[{{x}_{1}}={{x}_{1}}\]

\[{{x}_{2}}={{x}_{1}}+d\]

\[{{x}_{3}}={{x}_{1}}+2d\]

\[{{x}_{n}}={{x}_{1}}+(n-1)d\]

Adding \[{{S}_{n}}={{x}_{1}}+{{x}_{2}}+.....{{x}_{n}}=n({{x}_{1}})+(0+1+....(n-1)\times d\]

                        \[=n{{x}_{1}}+\frac{n(n-1)}{2}d\]

                        \[=\frac{n}{2}\left[ 2{{x}_{1}}+(n-1)d \right]\]

GEOMETRIC PROGRESSION (G.P.)

A sequence \[{{a}_{1}},{{a}_{2}},{{a}_{3}}.....,{{a}_{n}}\] is said to be in G.P., if the ratio of the consecutive terms is a constant, that is, \[\frac{{{a}_{2}}}{{{a}_{1}}}=\frac{{{a}_{3}}}{{{a}_{2}}}=r\]      \[\Rightarrow {{a}_{2}}={{a}_{1}}.r,\]

            \[{{a}_{3}}={{a}_{1}}{{r}^{2}}\]

Thus, if ‘r’ is the common ratio, then the nth term of the sequence is given by \[{{a}_{n}}=a{{r}^{n-1}}\]. The of n terms of the G.P. is given by.

\[{{S}_{n}}=\frac{a\left( {{r}^{n}}-1 \right)}{r-1};r>1\] and \[{{S}_{n}}=\frac{a\left( 1-{{r}^{n}} \right)}{1-r};r<1\]

Infinite G.P.: Sum of infinite G.P. is given by \[{{S}_{\infty }}=\frac{a}{1-r}\]

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

1. If each term of GP is multiplied or divided by a constant, then the resulting sequence is also in GP. For e.g. if G.P. \[a,ar.............a{{r}^{n-1}}\]is divided by k, then we get a new sequence, \[\frac{a}{k},\frac{ar}{k},....\frac{a{{r}^{n-1}}}{k}\]. This sequence is also m G.P.
2. When reciprocal of each term of a G.P. is taken, the resulting sequence is a G.P. For e.g., By taking reciprocal of above sequence, we get, \[\frac{1}{a},\frac{1}{ar},\frac{1}{a{{r}^{2}}},.......\frac{1}{a{{r}^{n-1}}}\]which can be re–written as,

\[\left( \frac{1}{a} \right).1,\left( \frac{1}{a} \right)\times \left( \frac{1}{r} \right),\left( \frac{1}{a} \right)\times \left( \frac{1}{{{r}^{2}}} \right),.....\left( \frac{1}{a} \right)\times \left( \frac{1}{{{r}^{n-1}}} \right)\] whose first term \[=\frac{1}{a}\] and common ratio \[=\left( \frac{1}{r} \right)\]

3. If each term of the GP is raised to the same power, then the resulting sequence is also in GP. For e.g., the above G.P. on squaring, will become \[{{a}^{2}},{{a}^{2}}{{r}^{2}},........{{a}^{2}}{{\left( {{r}^{n-1}} \right)}^{2}}\]

Further on G.P. 

1. While solving problems on geometric progression, we can take the three terms in geometric progression as a/r, a and ar. This automatically eliminates one of the unknowns by using the property of G.P.
2. If three terms are in geometric progression, then the middle term is the geometric mean of the G.P, i.e., if a, b and c are in G.P., then b is the geometric mean of the three terms.
3. If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},.....,{{a}_{n}}\] are in G.P. then the geometric mean (GM) of these n terms is given by \[=\sqrt[n]{{{a}_{1}}{{a}_{2}}{{a}_{3}}.....{{a}_{n}}}\].

HARMONIC PROGRESSION (H.P.)

A sequence is said to be in H.P. If the reciprocal of its terms are in A.P. Problem on HP are solved by taking reciprocal of terms and considering them to be in A.P.

Let three numbers a, b, c be in HP

Then, \[\frac{1}{a},\frac{1}{b}\]and\[\frac{1}{c}\] are in A.P. \[\Rightarrow \frac{1}{b}=\frac{1}{2}\left( \frac{1}{a}+\frac{1}{c} \right)\] which can also be written as, \[\frac{a}{c}=\frac{a-b}{b-c}\]

Harmonic Mean (HM)                                                       

If ‘a’ and ‘b’ be any two terms in HP, then their H.M. is given by H.M. \[=\frac{2ab}{a+b}\]

Relationship between AM, GM and HM

            \[AM=\frac{a+b}{2},GM=\sqrt{ab}\] and \[HM=\frac{2ab}{a+b}\]

            \[\Rightarrow AM\times HM=\frac{a+b}{2}\times \frac{2ab}{a\times b}=ab={{(GM)}^{2}}\]

            \[\left[ AM\times HM=G{{M}^{2}} \right]\,\Rightarrow \frac{AM}{GM}=\frac{GM}{HM}\] but \[\frac{AM}{GM}>1\therefore \frac{GM}{HM}>1\Rightarrow GM>HM\]

            Hence \[AM>GM>HM\].

            Also, \[AM-GM=\frac{a+b}{2}-\sqrt{ab}=\frac{a+b-2\sqrt{ab}}{2}=\frac{{{\left( \sqrt{a}-\sqrt{b} \right)}^{2}}}{{{\left( \sqrt{2} \right)}^{2}}}={{\left( \frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}} \right)}^{2}}\]   \[\]