Tag: geometric sequences

Quizzes (2) Questions (161)

Questions Related to geometric sequences

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

$\Rightarrow x = 4$

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

Say true or false.
The total savings (in $Rs.$) after every month for $10$ months when $Rs. 50$ are saved each month are $50, 150, 200, 250, 300, 350, 400, 450, 500$ represent G.P.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Ratio of second term to first term is $ \frac {150}{50} = 3$

Ratio of third term to second term is  $ \frac {200}{150} = 1.33$

Thus, the ratio is not matching. 

Hence, it is not a GP.

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

Say true or false.
Given series:
$15, 30, 60, 120, 240$ is in G.P.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given series is $15,30,60,120,240$
Ratio between first two terms $=$ $\dfrac{30}{15}$ $=2$
Ratio between second and third terms $=$ $\dfrac{60}{30}$ $=2$
Ratio between third and fourth terms $=$ $\dfrac{120}{60}$ $=2$
Ratio between fourth and fifth terms $=$ $\dfrac{240}{120}$ $=2$
Since, the ratio between the terms is the same. The series forms a G.P.

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

Which of the following is not a G.P.?

  1. $2, 4, 6, 8....$
  2. $5, 25, 125, 625....$
  3. $1.5, 3.0, 6.0, 12.0....$
  4. $8, 16, 24, 32, ....$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

In series $2,4,6,8,....$ difference is same i.e. $2$

In $8,16,24,32,......$ difference again is same $8$
$\therefore$ both the series (a) and (b) are in AP as the difference between their consecutive terms is the same.

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

For the infinite series $1-\cfrac { 1 }{ 2 } -\cfrac { 1 }{ 4 } +\cfrac { 1 }{ 8 } -\cfrac { 1 }{ 16 } -\cfrac { 1 }{ 32 } +\cfrac { 1 }{ 54 } -\cfrac { 1 }{ 128 } -....\quad $ let $S$ be the (limiting) sum. Then $S$ equals

  1. $0$
  2. $\cfrac { 2 }{ 7 } $
  3. $\cfrac { 6 }{ 7 } $
  4. $\cfrac { 9 }{ 32 } $
  5. $\cfrac { 27 }{ 32 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Combine the terms in threes, to get the geometric series
$\cfrac { 1 }{ 4 } +\cfrac { 1 }{ 32 } +\cfrac { 1 }{ 256 } +....;\quad \quad S=\cfrac { \cfrac { 1 }{ 4 }  }{ 1-\cfrac { 1 }{ 8 }  } =\cfrac { 2 }{ 7 } $ or
rearrange the terms into three series:
$1+\cfrac { 1 }{ 8 } +\cfrac { 1 }{ 64 } +...\quad -\cfrac { 1 }{ 2 } -\cfrac { 1 }{ 16 } -\cfrac { 1 }{ 128 } -....,\quad -\cfrac { 1 }{ 4 } -\cfrac { 1 }{ 32 } -\cfrac { 1 }{ 256 } -....\quad $
${ S } _{ 1 }=\cfrac { 1 }{ 1-\cfrac { 1 }{ 8 }  } =\cfrac { 8 }{ 7 } ;{ S } _{ 2 }=\cfrac { -\cfrac { 1 }{ 2 }  }{ 1-\cfrac { 1 }{ 8 }  } =-\cfrac { 4 }{ 7 } ;{ S } _{ 3}=\cfrac { -\cfrac { 1 }{ 4 }  }{ 1-\cfrac { 1 }{ 8 }  } =-\cfrac { 2 }{ 7 } ;\quad \therefore S=\cfrac { 2 }{ 7 } $