A set of numbers where the numbers are arranged in a definite order, like the natural number , is called a sequence. Some examples sequences are natural numbers , even integers between 10 and 100 , squares of integers , etc. Sequences that follow specific patterns are called progressions. There are arithmetic progression , geometric progression and harmonic progression according to patterns. In this article , we see different progressions , n th term of sequence and sum of n terms of sequence and some examples with solutions.
What is a sequence ?
In a general, a sequence is written as `t_1 ,t_2, t_3 , t_4,...... t_n `
where t_1 - $ first trem , $t_4 -$ fourth term $ ,........, t_n - n$ th term
Finite sequence - A sequence containing finite number of terms is called a finite sequence. It is written as $\{t_1 ,t_2, t_3 , t_4,...... t_n\}$ for some positive integer n .
Infinite sequence - A sequence is said to be infinite if it is not a finite sequence. It is written as $\{t_1 ,t_2, t_3 \} $ or $\{t_n\} n\geq 1$.
Sequences that follow specific patterns are called progressions.
Arithmetic Progression (A. P.)
Consider the following sequences,
1) 2,5,8,11,14,......
2)4,10,16,22,28,....
3)4,16,64,256,....
4)-3,2,7,12,17,...
The sequence 1,2 and 5 are A. P. but the term in sequences 4 are not in A.P. as the difference between consecutive term is not constant.
If $t_1,t_2,t_3,....t_n$ are in A.P. then $t_{n+1} -t_n=d$ , is constant for all n.
Hence the squence can also be wriiten as $a,a+d,a+2d,.....$'
It's $n$ th term is
$$t_n=a+(n-1)d , t_1 =a$$
Sum of $n$ terms,
$$S_n= t_1+t_2+......+t_n = \frac{n}{2} [2a+(n-1)d]$$
If $t_1 ,t_2, t_3 , t_4,...... t_n $ are in A.P. ,
i) $t_1+k , t_2+k, t_3+k , t_4+k ,...... t_n+k$ are also in A.P.
ii) $kt_1 , kt_2, kt_3 ,,...... kt_n $ are also in A.P.
Geometric Progression
Hence a G. P. can also be written as $a, ar^2, ar^3,.....$ where $a$ is first term and $r$ is common ratio.
Examples:
1) $2,4,6,8,16,.....[a=2, r=2]$
2) $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27,....} [a=1, r=\frac{1}{3}]$
3) $1,-1,1,-1,1,-1 ,....[a=1, r=-1]$
The General term or the n th term of a G.P.
If a and r are the first term and common ration of a G.P. respectively. Then its n th term is given by
$$t_n=ar^{n-1}$$
Properties of Geometric Progression
If $t_1, t_2,t_3,....t_n$ are in G.P. ,
i) $\frac{1}{t_1}, \frac{1}{t_2}, \frac{1}{t_3},......, \frac{1}{t_n} are also in G.P. $
ii) $(k \neq 0) kt_1, kt_2, kt_3,...kt_n$ are also in G.P.
iii) $t_1^n, t_2^n, t_3^n,... are also in G.P.
Sum of the first n terms of a G.P. $(S_n)$
Consider the G.P. $t_1, t_2,t_3,....t_n$ , we write the sum of first n terms,
$$t_1+t_2+t_3+.....+t_n as \sum_{r=1}^n t_r = S_n$$
Note : $\sum $ is notation of summation , the sum os of all $t_r ( 1 \leq r\leq n) $
In $\sum_{r=1}^n$ the variable is r.
Theorem: If $a, ar^2, ar^3,.....,ar^{n-1}$ is a G.P. then
$$S_n= a+ar^2+ar^3+...+ ar^{n-1}= t_1+t_2+t_3+.....+t_n = \sum_{r=1}^n t_r = \frac{a(1-r^n)}{(1-r)} (r \neq 1)$$
Let's Note:
$$S_n=\frac{a(1-r^n)}{(1-r)} (r \neq 1)$$
$$S_n=\frac{a(r^n)-1}{(r-1)} (r \neq 1)$$
3) If $r=1$, then G.P. is $a, a,a,....a$ (n times ) So, $S_n=a\cdot n$
4) $S_n-S_{n-1}=t_n$
Sum of infinite terms of G.P.
Consider a G.P. of positive terms. The sum of first $n$ terms is
$$\frac{a(r^n)-1}{(r-1)}=\frac{a(1-r^n)}{(1-r)}$$
If $|r| \gt 1, r-1$ is constant but $r^n$ approaches $\infty$ as n approaches $\infty$ , so the infinite terms cannot be summed up.
If $| r | \lt 1, r^n$ approaches 0, as n approaches $\infty$ and the sum $S_n=\frac{a(1-r^n)}{(1-r)}$ approaches $\frac{a}{1-r}$ . Hence the infinite sum $\sum _{r=1}^{\infty}t_r$ is said to be \frac{a}{1-r}.
Harmonic Progression ( H. P. )
Definition : A sequence $t_1, t_2,t_3,....t_n .....(t_n \neq 0, n \in N)$ is called a harmonic progression if $\frac{1}{t_1}, \frac{1}{t_2}, \frac{1}{t_3}, ......, \frac{1}{t_n}$ are in A.P.
Examples:
i) $\frac{1}{7}, \frac{1}{11}, \frac{1}{15}, .......$ are in H.P. as
$\frac{1}{(\frac{1}{7})}, \frac{1}{(\frac{1}{11})}, \frac{1}{(\frac{1}{15})} i.e. 7,11,15,.... are in A.P.$
Types of Means:
Arithmetic mean (A. M.):
If x and y are two numbers, their A.M. is given by
$$A=\frac{x+y}{2}$$
We observe the $x, A,y$ form an A.P.
$$A-x=y-A , \therefore 2A=x+y , \therefore A=\frac{x+y}{2}$$
Geometric mean (G. M.) :
If x and y are two numbers having same sign (positive or negative), their G.M. is given by
$$G= \sqrt{xy}$$
We observe the $x, G, y$ form a G.P.
$$\frac{G}{x}\frac{y}{G} , \therefore G^2=xy $$
$$G=\sqrt{xy}$$
Harmonic mean (H.M.) :
If x and y are two numbers, their H.M. is given by
$$H=\frac{2xy}{x+y}$$
We observe that $x, H, y$ form an H.P. i.e. $\frac{1}{x}, \frac{1}{H}, \frac{1}{y}$ is in A.P.
$$\frac{1}{H}=\frac{\frac{1}{x}+\frac{1}{y}}{2}=\frac{x+y}{2xy} , \therefore H=\frac{2xy}{x+y}$$
Note : 1) These results cane be extended to n numbers as follows:
$$A=\frac{x_1+x_2+x_3+......+x_n}{n}$$
$$G= \sqrt[n]{x_1.x_2.x_3.,....x_n}$$
$$H=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+.......+ \frac{1}{x_n}}$$
2) If $x=y$ then $A=G=H$
Theorem : If A, G and H are A.M. , G. M. , H.M. of two positive numbers x and y respectively, then
$i) G^2= AH ~~~ii) A \geq G \geq H $
Frequently Asked Questions – FAQ
1)What is a sequence ?
A set of numbers where the numbers are arranged in a definite order, like the natural number , is called a sequence. Some examples sequences are natural numbers , even integers between 10 and 100 , squares of integers , etc.
2) What are types of sequences ?
The types of sequence are:
i) Arithmetic sequence
ii) Geometric sequence
iii) Harmonic sequence
3) What is $n$ the term and sum of $n$ in arithmetic progression ?
In arithmetic progression,It's nth term is
$$t_n=a+(n-1)d , t_1 =a$$
Sum of n terms,
$$S_n= t_1+t_2+......+t_n = \frac{n}{2} [2a+(n-1)d]$$
4)What is geometric progression?
A sequence $t_1 ,t_2, t_3 , t_4,...... t_n $ is G.P. (Geometric Progression) if the common ratio $\frac{t_{n+1}}{t_n}=r$ is constant for all n.
5) What is harmonic progression?
A sequence $t_1, t_2,t_3,....t_n .....(t_n \neq 0, n \in N)$ is called a harmonic progression if $\frac{1}{t_1}, \frac{1}{t_2}, \frac{1}{t_3}, ......, \frac{1}{t_n}$ are in A.P.
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