Tag: understanding geometric progressions

Questions Related to understanding geometric progressions

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

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

  1. arithmetic sequence

  2. geometric sequence

  3. geometric series

  4. harmonic sequence

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Correct Option: B
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.

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

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

  1. geometric sequence

  2. arithmetic sequence

  3. geometric series

  4. harmonic sequence

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Correct Option: A
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 .

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: 

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

  1. geometric sequence

  2. arithmetic sequence

  3. geometric series

  4. harmonic sequence

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Correct Option: A
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.

Identify the geometric progression.

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

  1. $1, 3, 5, 7, 9, ...$

  2. $2, 4, 6, 8, 10...$

  3. $5, 10, 15, 25, 35..$

  4. $1, 3, 9, 27, 81...$

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Correct Option: D
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 $3$.

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:

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

  1. geometric series

  2. arithmetic progression

  3. harmonic sequence

  4. geometric sequence

Show answer
Correct Option: D
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 $(D)$

Which one of the following is not a geometric progression?

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

  1. $1, 2, 4, 8, 16, 32$

  2. $4, -4, 4, -4, 4$

  3. $12, 24, 36, 48$

  4. $6, 12, 24, 48$

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Correct Option: C
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.

Which one of the following is a geometric progression?

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

  1. $3, 5, 9, 11, 15$

  2. $4, -4, 4, -4, 4$

  3. $12, 24, 36, 48$

  4. $6, 12, 24, 36$

Show answer
Correct Option: B
Explanation:

$4, -4, 4, -4, 4$ is a geometric progression.
Here the common ratio is $-1$.

Option B is correct.

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

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

  1. $2 + 6 + 18 + 54 +...$

  2. $3 + 12 + 48 + 192 +....$

  3. $1 + 4 + 7 + 10 +....$

  4. $1 + 3 + 9 + 27 +....$

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Correct Option: C
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.

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

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

  1. $1, 1, 1, 1, 1$

  2. $1, 2, 3, 4, 5$

  3. $2, 4, 6, 8, 10$

  4. $-1, 2, -3, 4, -5$

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Correct Option: A
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 $1,1,1,1,1,............$.

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

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

  1. arithmetic progression

  2. harmonic progression

  3. geometric progression

  4. geometric series

Show answer
Correct Option: C
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.