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

$4, \dfrac{8}{3}, \dfrac{16}{9}, \dfrac{32}{27}..$ is a

  1. arithmetic sequence

  2. geometric sequence

  3. geometric series

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

lets check the ratio between the consecutive terms.
$\dfrac {\frac {8}{3}}{4}=\dfrac {8}{12}=\dfrac {2}{3}$
Again take the ratio between next consecutive terms.
$\dfrac {\frac {16}{9}}{\frac {8}{3}}=\dfrac {16\times 3}{9\times 8}=\dfrac {2}{3}$
Here the common ratio is same $\dfrac{2}{3}$ throughout.
Hence, $4, \dfrac{8}{3}, \dfrac{16}{9}, \dfrac{32}{27}..$ is a geometric sequence.

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. geometric sequence

  2. arithmetic sequence

  3. geometric series

  4. harmonic sequence
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Sol:
We know that if a,b,c are in G.p then $b^2=ac$
We know that a G.P
$a,ar,ar^2.ar^3-----ar^n$
${ a } _{ 1 }{ ,a } _{ 2 },{ a } _{ 3 },{ a } _{ 4 },----{ a } _{ n }$
$\dfrac { { a } _{ 2 } }{ { a } _{ 1 } } =\dfrac { { a } _{ 3 } }{ { a } _{ 2 } } =\dfrac { { a } _{ 3 } }{ { a } _{ 3 } } ----\dfrac { { a } _{ n } }{ { a } _{ n-1 } } =r$  (r=constant)
Therefore in a geometric progression each term is found multiplying the previous term by constant .
Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

If a sequence of values follows a pattern of multiplying a fixed amount times each term to arrive at the following term, it is called a: 

  1. geometric sequence

  2. arithmetic sequence

  3. geometric series

  4. harmonic sequence
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$3,3^2,3^3,3^4,....(r=3)$
In a sequence if a fixed amount/constant is multiplied to each term to get the successive term the sequence is called geometric sequence.
Here $3$ is the constant which gets multiplied to each term to obtain the successive term.
Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

Identify the geometric progression.

  1. $1, 3, 5, 7, 9, ...$
  2. $2, 4, 6, 8, 10...$
  3. $5, 10, 15, 25, 35..$
  4. $1, 3, 9, 27, 81...$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A geometric sequence 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.
So, $1, 3, 9, 27, 81...$ is a geometric progression.
Here the common ratio is $3$.

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

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 is known as:

  1. geometric series

  2. arithmetic progression

  3. harmonic sequence

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

A sequence of number, ${ a } _{ 1 }+{ a } _{ 2 }+......{ a } _{ n }$ quotient of any two successive number is a constant,

$\cfrac { { a } _{ 2 } }{ { a } _{ 1 } } =\cfrac { { a } _{ 3 } }{ { a } _{ 2 } } =........=\cfrac { { a } _{ n } }{ { a } _{ n-1 } } =$common ratio $(r)$
So we can write
${ a } _{ 1 }+{ a } _{ 1 }r+{ a } _{ 2 }r+{ a } _{ 3 }r.......{ a } _{ n-1 }r\ ={ a } _{ 1 }+{ a } _{ 1 }r+{ a } _{ 1 }{ r }^{ 2 }..........{ a } _{ n-2 }{ r }^{ 2 }$
and in the end in terms of ${ a } _{ 1 }$
$={ a } _{ 1 }+{ a } _{ 1 }r+{ a } _{ 1 }{ r }^{ 2 }+{ a } _{ 1 }{ r }^{ 3 }.........{ a } _{ 1 }{ r }^{ n-1 }$
We can clearly say this series is in $GP$.
Answer $(D)$

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

Which one of the following is not a geometric progression?

  1. $1, 2, 4, 8, 16, 32$
  2. $4, -4, 4, -4, 4$
  3. $12, 24, 36, 48$
  4. $6, 12, 24, 48$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
For geometric progression, the ratio of the consecutive terms should be equal.
Here $12, 24, 36, 48$ is not a geometric progression. Here only the difference is common i.e. $12$.
Rest all options have same common ratio i.e., in option A, the ratio is $2$. In option B, the ratio is $-1$.
And in option D, the ratio is $2$.
Here the given sequence is an arithmetic progression.
Multiple choice maths geometric sequences introduction to geometric progression understanding geometric progressions introduction to geometric progressions

Which of the following is not in the form of G.P.?

  1. $2 + 6 + 18 + 54 +...$
  2. $3 + 12 + 48 + 192 +....$
  3. $1 + 4 + 7 + 10 +....$
  4. $1 + 3 + 9 + 27 +....$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In option A, the common ratio is $3$.
In option B, the common ratio is $4$.
In option D, the common ratio is $3$.
$1 + 4 + 7 + 10 +...$. is not a G.P., since the sequence is in the form of A.P.

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

Which one of the following is a general form of geometric progression?

  1. $1, 1, 1, 1, 1$
  2. $1, 2, 3, 4, 5$
  3. $2, 4, 6, 8, 10$
  4. $-1, 2, -3, 4, -5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Lets see option A:
Sequence is $1,1,1,1,1$
General form of GP is $a=1$ and $r=1$
Here ratio is constant throughout.
Thus in all options, option A is correct.
The geometric progression is $1,1,1,1,1,............$.

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

The number of terms in a sequence $6, 12, 24, ....1536$ represents a

  1. arithmetic progression

  2. harmonic progression

  3. geometric progression

  4. geometric series
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Given series is $6,12,24,....1536$
Since, $\dfrac {12}{6} =2$ and $\dfrac {24}{12} =2$
i.e. the given sequence is a geometric sequence / progression.