Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Geometric Progression (G.P.)

If    \dfrac{t_{n+1}}{t_n}=r (constant), for  n=1,2,3,... then  (t_n) is called a Geometric Sequence or Geometric Progression (G.P). and the series  \sum_{n=1}^{\infty }t_n is called a geometric series. The constant  r is known as the common ratio (c.r) .

Note -:

(i) If  t_1\text{( first term )}=a , common ratio  =r , then  t_n = ar^{n-1}

(ii) No term of a GP can be zero, for otherwise  \dfrac{t_{n+1}}{t_n} will be meaningless for the corresponding value of  n .

 n^{th} Partial Sum of a Geometric Series -:

For a geometric series with  t_n=a and common ratio ratio  =r ,  S_n=t_1+t_2+...+t_{n-1}=a+ar+ar^2+...+ar^{n-1}

 \therefore rS_n=ar+ar^2+...+ar^{n-1}+ar^n

 \implies \text{(Subtracting)}(1-r)S_n = a (1-r^n)

 \therefore S_n=\dfrac{a(1-r^n)}{1-r} , for    r\neq 1

If  r=1 , then  t_n=a , for every  n , so that  S_n=na

Sum of Geometric Series -:

If  |r|  <  1 , i.e.  -1 <  r <  1 then  r^n\to 0 . When  n\to \infty  . So for the geometric series with  |r|  <  1 .

We have  \lim_{n \to \infty } S_n=\lim_{n \to \infty } \ a \cdot\dfrac{1-r^n}{1-r}=\dfrac{a}{1-r}

Therefore  \sum_{n=1}^{\infty }ar^{n-1}=\dfrac{a}{1-r} ; if  \left| r \right| <  1 and  \sum_{n=1}^{\infty }ar^{n-1} diverges if  \left| r \right|> 1 .

Hence the geometric series  \sum_{n=1}^{\infty }ar^{n-1} converges if  \left| r \right| <  1  and diverges if  \left| r \right| >1 .

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