Tag: introduction to geometric progressions

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

If $\alpha, \beta, \gamma$ are non-constant terms in G.P and equations $\alpha { x }^{ 2 }+2\beta x+\gamma =0\quad $ and ${x}^{2}+x-1=0$ has a common root then $\left( \gamma -\alpha  \right) ,\beta $ is

  1. $\alpha \beta $
  2. $\beta \gamma $
  3. $\gamma \alpha $
  4. $0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the common ratio of G.P is $r$ Therefore $\quad \beta =\alpha t,\alpha { t }^{ 2 }$
Equation $\alpha { x }^{ 2 }+2\alpha rx+\alpha { t }=0\quad 
\Rightarrow { x }^{ 2 }+2rx+{ t }^{ 2 }=0....(i)$
Given equation (i) and ${ x }^{ 2 }+x-1=0....(ii)$ has a common root
$(i)-(ii)\Rightarrow (2e-1)x+({ r }^{ 2 }+1)=0\Rightarrow x=\cfrac { -\left( { r }^{ 2 }+1 \right)  }{ 2r-1 } ....(iii)\quad $
Putting (iii) in equation (ii) $\Rightarrow { \left( { r }^{ 2 }+1 \right)  }^{ 2 }-\left( { r }^{ 2 }+1 \right) (2r-1)-{ \left( { 2r }^{ 2 }-1 \right)  }^{ 2 }=0\Rightarrow { r }^{ 4 }-2{ r }^{ 3 }-{ r }^{ 2 }+2r+1=0....(iv)$
dividing equation (iv) by ${r}^{2}$ $\Rightarrow { \left( r-\cfrac { 1 }{ r }  \right)  }^{ 2 }-2{ \left( r-\cfrac { 1 }{ r }  \right)  }+1=0\Rightarrow { \left( r-\cfrac { 1 }{ r } -1 \right)  }^{ 2 }=0\Rightarrow \cfrac { r-1 }{ r } =1....(v)\quad $
$\left( \gamma -\alpha  \right) \beta =\left( \alpha { r }^{ 2 }-\alpha  \right) \times \alpha r={ \alpha  }^{ 2 }\left( { \alpha  }^{ 2 }-1 \right) r={ \alpha  }^{ 2 }(r-1)={ \alpha  }^{ 2 }{ r }^{ 2 }$
(using $(v)=\alpha \times \alpha { t }^{ 2 }\quad $

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

Write down the first five terms of the geometric progression which has first term 1 and common ratio 4.

  1. 1, 4, 16, 64, 244

  2. 1, 4, 24, 64, 256

  3. 1, 4, 16, 32, 256

  4. 1, 4, 16, 64, 256
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let a and d be the first term and common ratio of the GP respectively.
Given a=1 and d=4.
Now, $a _n=ar^{n-1}$
$\therefore a _1=a=1$
$a _2=ar=1\times4=4$
$a _3=ar^2=1\times(4)^2=16$
$a _4=ar^3=1\times(4)^3=64$
$a _5=ar^4=1\times(4)^4=256$





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

$\displaystyle \frac{1}{c},(\frac{1}{ca})^{\dfrac{1}{2}},\frac{1}{a}$ is in

  1. AP

  2. GP

  3. HP

  4. NONE
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given series


$\dfrac{1}{c},\left(\dfrac{1}{ca}\right)^{\dfrac{1}{2}},\dfrac{1}{a}$

Lets consider a G.P of elements $A,B,C$

 $\therefore$ Geo.mean $\Rightarrow B^2=AC$

Comparing it with given series.

$A=\dfrac{1}{c}B=\left(\dfrac{1}{ca}\right)^{\dfrac{1}{2}},C=\dfrac{1}{a}$

$\therefore B^2=\left(\dfrac{1}{ca}\right)^{\dfrac{1}{2}\times 2}$

            $=\dfrac{1}{ca}$........(1)

$AC=\dfrac{1}{c}\times \dfrac{1}{a}=\dfrac{1}{ca}$..............(ii)

$\therefore (i)=(ii)$

$\therefore B^2=AC$ So given series is in G.P 

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

Find the sum the infinite G.P.: $\displaystyle {\frac{2}{3}\, -\, \frac{4}{9}\, +\, \frac{8}{27}\, -\, \frac{16}{21}\, +\, ........}$ 

  1. $\displaystyle \frac{2}{5}$
  2. $\displaystyle \frac{3}{5}$
  3. $\displaystyle \frac{19}{27}$
  4. $\displaystyle \frac{8}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know, $S _{\infty }=\dfrac{a}{1-r}$
From the given series, $a=\dfrac{2}{3} ,r=-\dfrac{2}{3}$
$\therefore S _{\infty }=\dfrac{\frac{2}{3}}{1-\left ( -\frac{2}{3} \right )}$


$\Rightarrow \dfrac{\frac{2}{3}}{1+\frac{2}{3}}$

$\Rightarrow \dfrac{\frac{2}{3}}{\frac{5}{3}}$

$\Rightarrow \dfrac{2}{5}$

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

Sum the series: $\displaystyle {1\, -\, \frac{1}{3}\, +\, \frac{1}{3^2}\, -\, \frac{1}{3^3}\, +\, \frac{1}{3^4}.......\infty}$

  1. $\displaystyle \frac{3}{4}$
  2. $\displaystyle \frac{4}{3}$
  3. $\displaystyle \frac{2}{3}$
  4. $\displaystyle \frac{1}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know, $S _{\infty }=\dfrac{a}{1-r}$
From the given series, $a=1 ,r=-\dfrac{1}{3}$
$\therefore S _{\infty }=\dfrac{1}{1-\left ( -\dfrac{1}{3} \right )}$
$= \dfrac{1}{1+\dfrac{1}{3}}$


$ =\dfrac{1}{\dfrac{4}{3}}$

$= \dfrac{3}{4}$

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

If a, b and c are in geometric progression, then $a^2$, $b^2$ and $c^2$ are in _____ progression.

  1. AP

  2. GP

  3. HP

  4. AGP
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $ a,b,c $ are in GP.
So, the common ratio between the first and second term ; second and third will be the same.
$ => \dfrac {b}{a} = \dfrac {c}{b} $


$ => b^2 = ac $

If we square both sides, we see that
$ (b^2)^2 = a^2 \times c^2 $

This means, even, $ a^2, b^2, c^2 $ are also in GP.

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

The sequence $-6 + 42 - 294 + 2058$ is a

  1. finite geometric sequence

  2. finite arithmetic sequence

  3. infinite geometric sequence

  4. infinite harmonic sequence
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sequence $-6 + 42 - 294 + 2058$ is a finite geometric sequence.
Here the common ratio is $-7$.

As it has finite terms, therefore the series is finite sequence.

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

The sum of the series $10 - 5 + 2.5 - 1.25.....$ is called

  1. finite geometric sequence

  2. finite arithmetic sequence

  3. infinite geometric sequence

  4. infinite harmonic sequence
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given series is $10-5+2.5-1.25.....$

Here the common ratio is $\dfrac {-5}{10}=-\dfrac {1}{2}$.
It is also never ending and continued.
Hence, the given series is infinite geometric series.

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

When a number $x$ is subtracted from each of the numbers $8, 16$, and $40$, the resulting three numbers form a geometric progression. Find the value of $x$.

  1. $3$
  2. $4$
  3. $6$
  4. $12$
  5. $18$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that ${(16-x)}^{2}=(8-x)(40-x)$
$\Rightarrow 256-32x+{x}^{2} = 320-48x+{x}^{2}$
$\Rightarrow 16x = 64$ 

$\Rightarrow x = 4$