Tag: introduction to geometric progression

Quizzes (0) Questions (87)

Questions Related to introduction to geometric progression

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

A progression of the form $a, ar, ar^2$, ..... is a

  1. geometric series

  2. harmonic series

  3. arithmetic progression

  4. geometric progression
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A progression of the form $a, ar, ar^2$, ..... is a geometric progression.
Geometric Progression refers to a sequence in which successor term of each term is obtained by multiplying a constant term.

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

The geometric progression which have infinite terms is called

  1. finite geometric progression

  2. finite arithmetic progression

  3. infinite geometric progression

  4. finite harmonic progression
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The geometric progression which have infinite terms is called infinite geometric progression.
$1 + 0.5 + 0.25 + 0.125....$ is an example of infinite geometric progression.

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

The sum $1+\dfrac { 2 }{ x } +\dfrac { 4 }{ { x }^{ 2 } } +\dfrac { 8 }{ { x }^{ 3 } } +....\left( up\ to\ \infty  \right) ,x\neq 0,$ is finite if

  1. $\left| x \right| < 2$
  2. $\left| x \right| > 2$
  3. $\left| x \right| < 1$
  4. $2\left| x \right| < 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$1 + \dfrac{2}{x} + \dfrac{4}{{{x^3}}} + \dfrac{8}{{{x^3}}}.......\,upto\,\,\infty $
It is infinite $G.P$ with Common ratio $r=\dfrac{2}{x}$
It's sum is infinite if $\left| r \right| < 1$
i-e    
 $\left| {\dfrac{2}{x}} \right| < 1$
 $ \Rightarrow \dfrac{2}{{\left| x \right|}} < 1$
 $ \Rightarrow 2 < \left| x \right|$
$ \Rightarrow \left| x \right| > 2$       
Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

The sum of the infinite series $1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+......$

  1. Cannot be determined.
  2. Equals $\dfrac{15}{8}$
  3. Equals $2$
  4. Will be higher than $2$.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$Given\>series\>is\>an\>infinite\>GP\>with\>first\>term\>=1\>and\>common\>ratio=1/2\\\therefore\>sum=(\frac{a}{1-r})\\=(\frac{a}{1-1/2})=2$

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

$S = {3^{10}} + {3^9} + \frac{{{3^9}}}{4} + \frac{{{3^7}}}{2} + \frac{{{{5.3}^6}}}{{16}} + \frac{{{3^2}}}{{16}} + \frac{{{{7.3}^4}}}{{64}} + .........$ upto infinite terms, then $\left( {\frac{{25}}{{36}}} \right)S$ equal to 

  1. ${6^9}$
  2. ${3^{10}}$
  3. ${3^{11}}$
  4. ${2.3^{10}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\begin{array}{l}S = {3^{10}} + {3^9} + \cfrac{{{3^9}}}{4} + \cfrac{{{3^7}}}{2} + \cfrac{{{{5.3}^6}}}{{16}} + \cfrac{{{3^2}}}{{16}} + \cfrac{{{{7.3}^4}}}{{64}} + .........\infty \S = \cfrac{{1 \times {3^{10}}}}{{{2^0}}} + \cfrac{{2 \times {3^9}}}{{{2^1}}} + \cfrac{{3 \times {3^8}}}{{{2^2}}} + \cfrac{{4 \times {3^7}}}{{{2^3}}} + \cfrac{{5 \times {3^6}}}{{{2^4}}} + \cfrac{{6 \times {3^5}}}{{{2^5}}} + \cfrac{{7 \times {3^4}}}{{{2^6}}} + .....\infty \\cfrac{S}{6} = \cfrac{{{3^9}}}{2} + \cfrac{{2 \times {3^8}}}{{{2^2}}} + .......\infty \S - \cfrac{S}{6} = \cfrac{{{3^{10}}}}{2^0} + \cfrac{{{3^9}}}{{{2^1}}} + \cfrac{{{3^8}}}{{{2^2}}} + ........\infty \\cfrac{{6S - S}}{6} = \cfrac{{{3^{10}}}}{{1 - \cfrac{1}{6}}} = \cfrac{{{3^{10}}}}{5}\left( 6 \right) \end{array}$

$\dfrac56S=\dfrac{3^{10}\times 6 }{5}$
$\therefore \dfrac {25}{36}S=3^{10}$

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

In each of the following questions, a series of number is given which follow certain rules. One of the number is missing. Choose the missing number from the alternatives given below and mark it on your answer-sheet as directed. $1, \dfrac {1}{3}, \dfrac {1}{9}, \dfrac {1}{27}, \dfrac {1}{81}, \dfrac {1}{243}, $?

  1. $\dfrac {1}{729}$
  2. $\dfrac {1}{829}$
  3. $\dfrac {1}{749}$
  4. $\dfrac {1}{769}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

G.P series with common ratio $\dfrac 13$

$\therefore$, next term is $\dfrac {1}{243} \times \dfrac {1}{3}=\dfrac {1}{729}$

Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

Find the sum of an infinite G.P : $\displaystyle 1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+.......$

  1. $\displaystyle \frac{3}{5}$
  2. $\displaystyle \frac{3}{2}$
  3. $\displaystyle \frac{49}{27}$
  4. $\displaystyle \frac{8}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given series is $1+ \dfrac {1}{3}+ \dfrac {1}{9 }+ \dfrac {1}{27}+......$

$a=1, r= \dfrac {1}{3}$

$\therefore S _{\infty}=\dfrac{a}{1-r}$

$S _{\infty}=\dfrac{1}{1-\dfrac{1}{3}}$

$S _{\infty}=\dfrac{1}{\dfrac{2}{3}}$

$\therefore S _{\infty}=\dfrac{3}{2}$