Tag: understanding geometric progressions

Questions Related to understanding 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.

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

  1. True

  2. False

Show answer
Correct Option: B
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.

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

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

  1. True

  2. False

Show answer
Correct Option: A
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.

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

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

  1. $2, 4, 6, 8....$

  2. $5, 25, 125, 625....$

  3. $1.5, 3.0, 6.0, 12.0....$

  4. $8, 16, 24, 32, ....$

Show answer
Correct Option: A,D
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.

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

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

  1. $0$

  2. $\cfrac { 2 }{ 7 } $

  3. $\cfrac { 6 }{ 7 } $

  4. $\cfrac { 9 }{ 32 } $

  5. $\cfrac { 27 }{ 32 } $

Show answer
Correct Option: B
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 } $

What is the geometric mean of $6$ and $24$ ?

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

  1. $10$

  2. $12$

  3. $14$

  4. $16$

Show answer
Correct Option: B
Explanation:
Assume $a, b, c$ in G.P.
then $\dfrac{b}{a}=\dfrac{c}{b}$

$\Rightarrow b^2=ac$

$\Rightarrow b=\sqrt{ac}$

so, $G.M=\sqrt{ac}$

$=\sqrt{6\times 24}$

$G.M =12$

$a^x=b, b^y=c, c^z=a$
Find the value of x, y, z.

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

  1. $1$

  2. $Not$  $valid$

  3. $-1$

  4. $0$

Show answer
Correct Option: A
Explanation:

${ a }^{ x }=b\ { b }^{ y }=c=>{ a }^{ xy }=c\ { c }^{ z }=a\ { c }^{ z }=a=>{ b }^{ yz }=a\ \left( { a }^{ x } \right) ^{ yz }=a\ { a }^{ xyz }=a\ xyz=1$

If there exists a geometric progression containing 27, 8 and 12 as three of its terms (not necessarily consecutive) then no. of progressions possible are

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

  1. $1$

  2. $2$

  3. infinite

  4. None of these

Show answer
Correct Option: C
Explanation:
$8, 12, 29$

Let the first term be $8$

$\therefore a = 8$    ...(i)

Let the $p^{th}$ ther be $12$

$\therefore ar^{p-1} = 12$     ...(ii)

and $q^{th}$ term be $27$

$\therefore ar^{q-1} = 27$    ...(iii)

(i) / (ii)

$\dfrac{a}{ar^{p-1}} = \dfrac{8}{12}$

$\Rightarrow r^{p-1} = \dfrac{3}{2} = 1.5$      ...(iv)

(iii) / (i)

$r^{q-1} = \dfrac{27}{8} = \left(\dfrac{3}{2}\right)^3$

$r^{q-1} = (1.5)^3$    ...(v)

(v) / (iv)

$\Rightarrow \dfrac{r^{q-1}}{r^{p-1}} = \dfrac{(1.5)^3}{1.5}$

$\Rightarrow r^{q-1-p+1} = (1.5)^2$

$\Rightarrow r^{q-p} = (1.5)^2$

              $= (r^{p-1})^2$     ...from (iv)

$\Rightarrow r^{q-p} = r^{2p-2}$

$\therefore q-p = 2p-2$

$\Rightarrow \boxed{q=3p-2}$

for every distinct value of '$p$' there will be a district integer value of '$q$'

$\therefore $ Infinite no. of progression are possible.

option $C$