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Geometric sequences!!

Mohit_C

How do you find the number of terms in the following geometric sequence,
729, -243, 81....-1/3

Pls explain,
thanks

Reply 1

Simba

Mohit_C

How do you find the number of terms in the following geometric sequence,
729, -243, 81....-1/3

Pls explain,
thanks

Let the first term be 'a', the common ratio be 'r', and the number of terms be 'n'.

a = 729
r = -1/3

The last term = ar^{n - 1}.

-1/3 = 729*(-1/3)^{n - 1}

-1/2187 = (-1/3)^{n - 1}

Solving this gives n = 8.

Therefore, there are 8 terms in the sequence :smile:

...

Hope this helps,

~~Simba

Reply 2

KAISER_MOLE

consider what is happening with each term;

first term to second term occurs through division by -3, the same generates the next term

=> the common ratio = -1/3

we need to find when arⁿ = -1/3

Using a=729

=> rⁿ = -1/2187

this is the same as (-1/3)⁷

=> sequence goes from ar⁰ -> ar⁷ , and so contains 8 terms

Reply 3

yusufu

alternatively, type 729 in your calculator and press =
then type ANS/-3
729 is your first term. count how many times you press "=" until you get to -1/3.
:biggrin:

Reply 4

Mohit_C

Ok...how did you get n=8 from

-1/2187=-1/3^(n-1)?

Coz i tried using log but didnt work. Explain.

Reply 5

yusufu

Mohit_C

Ok...how did you get n=8 from

-1/2187=-1/3^(n-1)?

Coz i tried using log but didnt work. Explain.

-1/2187 = -1/3^n-1

=> 2187 = 3^n-1
take logs of this and solve for n

Reply 6

Mohit_C

Undry1

-1/2187 = -1/3^n-1

=> 2187 = 3^n-1
take logs of this and solve for n

Ok...looking at that i assume that u divided both sides by -1 rite? or else, how did you do that just get rid of the -1/...?

Reply 7

yusufu

-1/2187 = (-1)^n-1/3^n-1
multiplying by -1 gives
1/2187 = (-1)ⁿ/3^n-1
by inspection you can see that (-1)^n must be positive, giving
1/2187 = 1/3^n-1
=> 2187 = 3^n-1

Reply 8

Mohit_C

Undry1

-1/2187 = (-1)^n-1/3^n-1
multiplying by -1 gives
1/2187 = (-1)ⁿ/3^n-1
by inspection you can see that (-1)^n must be positive, giving
1/2187 = 1/3^n-1
=> 2187 = 3^n-1

I dont get this, how do you get rid of the - and get a positive number. Pls explain.

Reply 9

yusufu

Mohit_C

I dont get this, how do you get rid of the - and get a positive number. Pls explain.

after multiplying by -1
-1 * -1/2187 = -1 * (-1)^n-1/3^n-1

the left hand side becomes positive, therefore the right hand side must also be positive:
-1 * (-1)^n-1/3^n-1 is positive

now 3^n-1 is always positive no matter what value n takes
so -1 * (-1)^n-1 , i.e. (-1)ⁿ must be positive. the only way this is possible is if n is even, giving (-1)ⁿ=1

Reply 10

Mohit_C

Ok....finally got it. Thanks m8!

Reply 11

yusufu

good stuff. :smile:

Reply 12

Mohit_C

Ok i got another one here:

C2 Examination style:

A father promises his daughter an eternal gift on her birthday. On day 1 she receives €75 and each following day she receives 2/3 of the amount given to her the day before. The father promises that this will go on for ever. After k days the total amount of money that the daughter will have received exceeds €200. c) Find the smallest value of k.

Thanks.

Reply 13

yusufu

a= 75, r=2/3
75(1-(2/3)^k) / (1/3) > 200
1-(2/3)^k > 200/225
(2/3)^k < 1-(200/225)
klog(2/3) < log(1-(200/225))
k > 5.4 (inequality switches around because log(2/3) is negative)
k=6

Reply 14

Mohit_C

Undry1

a= 75, r=2/3
75(1-(2/3)^k) / (1/3) > 200
1-(2/3)^k > 200/225
(2/3)^k < 1-(200/225)
klog(2/3) < log(1-(200/225))
k > 5.4 (inequality switches around because log(2/3) is negative)
k=6

thanks

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