Question

19. The total number of lotuses in a lake grows in such a way that their number is increases
twice the previous day. For e.g. If there are x lotuses on 1" day, then on 2 day there will be
2x lotuses, 3 day there will be (2 x 2x = 4x) lotuses and so on. If there are 255 lotuses on
5 day, find the total number of lotuses on 7h day.
.​

Answer 1

Question:

The total number of lotuses in a lake grows in such a way that their number increases twice the previous day. For e.g. If there are x lotuses on 1st day, then on 2nd day there will be 2x lotuses, 3rd day there will be (2 x 2x = 4x) lotuses and so on. If there are 256 lotuses on 5th day, find the total number of lotuses on 7th day.

.

Answer:

1024 lotuses

Note:

• A sequence in which the ratio of consecutive terms are equal, then it is said to be in Geometric progression(GP).

• The general form of Geometric progression is given as : a , a•r² , a•r³ , a•r⁴..... , where "a" is the first term and "r" is the common ratio of the geometric progression(GP).

• The nth term of a geometric progression is given as ; T(n) = a•r^(n-1).

• The common ratio of a geometric progression is given by ; r = T(n)/T(n-1) .

Solution:

It is given that,

The total number of lotuses in a lake grows in such a way that their number increases twice the previous day.

If the number of lotuses on 1st day is x ,

Then the number of lotuses on 2nd day is 2x

Similarly,

No. of lotuses on 3rd day is 2•2x ie, 4x.

No. of lotuses on 4th day is 2•4x ie, 8x.

Clearly,

The number of lotuses is in Geometric progression.

Also,

The first term (no. of lotuses on 1st day) = x

And

The common ratio, r = T(n)/T(n-1)

= T(2)/T(2-1) { if n = 2 }

= T(2)/T(1)

= 2x/x

= 2

Also,

nth term of GP is given as , T(n) = {2^(n-1)}x

Thus,

The no. of lotuses on 5th day will be given as ;

=> T(5) = {2^(5-1)}x

=> 256 = 2⁴x

=> x = 256/2⁴

=> x = 256/16

=> x = 16

Now,

The no. of lotuses on 7th day will be given as ;

=> T(7) = {2^(7-1)}x

=> T(7) = (2^6)x

=> T(7) = 64•16

=> T(7) = 1024

Hence,

There would be 1024 lotuses in the lake on the 7th day.

Answer 2

${\boxed{\mathtt{\green{Question}}}}$

The total number of lotus in a lake grows in such a way that their number is increased twice the previous day if there are x lotuses on first day then 2x on the second and 4x on the third day. So if there are total 256 lotus is on 5th day find the total number of lotus is on 7th day.

${\boxed{\mathtt{\green{GIVEN}}}}$

• G.P = x , 2x , 4x .......
• Lotus on 5th day = 256

${\boxed{\mathtt{\green{To\: Find}}}}$

• Number of lotus flower on 7th day .

${\boxed{\mathtt{\green{Solution}}}}$

⇢ On reading the Question we got to know that the concept used in this is Geometric progression .

⇢ G.P = x , 2x , 4x , 8x, .........

$= \: r \: = \frac{a_{n + 1}}{a_n}$

$⇢ \: Common \: ratio \: = \frac{2x}{x} = 2$

${\boxed{\mathtt{\green{Formula}}}}$

$\boxed{a_{nth \: term} \: = a. {r}^{n - 1} }$

⇢ Number of lotus flower on 5th day =. 256

$= \: a_5 \: = a. {r}^{(5 - 1)}$

$= \: 256 \: = \: x. {(2)}^{4}$

$= \: 256 \: = 16x$

$= \: \frac{256}{16} = x$

$\boxed{x \: = \: 16}$

Now number of Lotus flowers on 7th day will be

$= \: a_7 \: = \: a. {r}^{(7 - 1)}$

$= \: a_7 \: = \: 16( {2)}^{6}$

$= \: a_7 \: = \: 16 \times 64$

${ \boxed{ \red{= \: a_7 \: = \: 1024}}}$

So , number of lotus flower on 7th day will be 1024 .