Tag: understanding geometric progressions

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

Find the GP whose $5^{th}$ term is $48$ and $9^{th}$ term is$ 768$.

  1. $3,6,12,24$
  2. $2,4,8,16$
  3. $6,12,24,48$
  4. $12,24,36,48$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle { ar }^{ 4 }=48$
$\displaystyle { ar }^{ 8 }=768$
$\displaystyle \therefore \quad { r }^{ 4 }=16$
$\displaystyle \therefore \quad r=2$
$\displaystyle a.{ 2 }^{ 4 }=48$
or, $\displaystyle a=\frac { 48 }{ 16 } =3$
The GP is 3,6, 12,24,.....

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

The reciprocals of all the terms of a geometric progression form a ________ progression.

  1. AP

  2. HP

  3. GP

  4. AGP
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let  $ a $ be the first term  and $ r $ be the common ratio of the GP. 

So, the series is $ a, ar, ar^2... $

Their reciprocals are $ \dfrac {1}{a}, \dfrac {1}{ar}, \dfrac {1}{ar^2} .. $

It is also a GP, with first term $ \dfrac {1}{a} $ and common ratio $ \dfrac {1}{r} $
Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

In a _______ each term is found by multiplying the previous term by a constant.

  1. arithmetic sequence

  2. geometric series

  3. arithmetic series

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

A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation index .

Or,
In a Geometric series each term is found by multiplying the previous term by a constant.
$(Ans \to B)$

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

A _________ is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a unchanging number called the common ratio.

  1. geometric progression

  2. arithmetic series

  3. arithmetic progression

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

A geometric progression is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a with a unchanging number called the common ratio.
Example: $2, 6, 18, 54, 108....$
This geometric sequence has a common ratio $3$.

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

$10,20,40,80$ is an example of

  1. fibonacci sequence

  2. harmonic sequence

  3. arithmetic sequence

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

$10, 20, 40, 80$ is an example of geometric sequence.
In geometric sequence, the ratio of succeeding term to the preceeding term is always equal.

Here the common ratio is $2$.

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

Identify the geometric series.

  1. $1 + 3 + 5 + 7 +....$
  2. $2 + 12 + 72 + 432...$
  3. $2 + 3 + 4 + 5 +...$
  4. $11 + 22 + 33 + 44+...$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Geometric series is of the following form:

$a+ar+ar^2+ar^3 +ar^4+..........+ar^n$
Series $2+12+72+432+......$ follows the same with $a=2$ and $r=6$.
Hence, option B is correct.

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

$1, 3, 9, 27, 81$ is a

  1. geometric sequence

  2. arithmetic progression

  3. harmonic sequence

  4. geometric series
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$1, 3, 9, 27, 81$ is a geometric sequence.
A geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.