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Questions Related to maths

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

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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. $6, 12, 24, 48$

Show answer
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. $1 + 3 + 9 + 27 +....$

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

Show answer
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. 

Find out the general form of geometric progression.

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

  1. $2, 4, 8, 16$

  2. $2, -2, 2, 3, 1$

  3. $0, 3, 6, 9, 12$

  4. $10, 20, 30, 40$

Show answer
Correct Option: A
Explanation:

The general form of geometric progression is $2, 4, 8, 16$.

Because here common ration between the consecutive terms is same. That is illustrated below.
$\dfrac {4}{2}=2, \dfrac {8}{4}=2, \dfrac {16}{8}=2$
Here the common ratio is $2$.

For which sequence below can we use the formula for the general term of a geometric sequence?

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. $4, 8, 16, 32, 64....$

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

Show answer
Correct Option: C
Explanation:

For a G.P., the ratio must be common throughout.
We use the formula for the general term of a geometric sequence for $4, 8, 16, 32, 64.... $
Here the common ratio is $2$.

An example of G.P. is

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

  1. $-1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}...$

  2. $ -1, \dfrac{3}{2}, \dfrac{1}{2}, -\dfrac{1}{2}$

  3. $1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{6}...$

  4. $1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}...$

Show answer
Correct Option: D
Explanation:

For a G.P., the ratio must be equal.
Here only D satisfies thiss condition.
So, an example of G.P. is $1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}...$
Here the common ratio is $\dfrac{1}{2}$.

The common ratio is used in _____ progression.

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

  1. arithmetic

  2. geometric

  3. harmonic

  4. series

Show answer
Correct Option: B
Explanation:

The common ratio is used in geometric progression.

For example: $2,4,8,16,....$
Here the common ratio is $2$.