Question

the first term of a G.P (Geometric Progression or Sequence) is 3 and the nth term is 192 . If the sum of the first n terms is 381 , find the constant ratio and the number of terms .

Answer 1

Okay, let's list all the things we know for this question.
SO the first term "a" is 3.
\[a=3\]
And the nth term is 192.
\[T_{n}=192\]
And the sum of the firs n terms is 381
\[S_{n}=381\]
\[T_{n}=ar^{n-1}\]
\[192=3r^{n-1}\]
\[64=r^{n-1}\]
\[S_{n}=\frac{a(r^n-1)}{r-1}\]
where r>1
\[S_{n}=\frac{a(1-r^n)}{1-r}\]
where r<1\
We will use the formula when r>1 first.
\[381=\frac{3(r^n-1)}{r-1}\]
\[127=\frac{r^n-1}{r-1}\]
\[127r-127=r^n-1\]
\[127r-r^n=126\]
\[r(127-r^{n-1}=126\]
I see something. Where have I seen r^{n-1} before???hmmmmmmmmmm.....That's right, r^{n-1}=64!!!!
So we sub that into the equation.
\[r(127-64)=126\]
Solve for r now please.

Answer 2

And tell me what you get.

Answer 3

r=2 ?

Answer 4

Do you really need to ask me whether it's right or wrong? I will verify for you this time only but later on you need to be confident. No need for the question mark. Yes it is correct.

Answer 5

Now find n for me.

Answer 6

Now you know what r is, find n.

Answer 7

You will need to use log here.

Answer 8

How ? I really don't know >3<

Answer 9

Okay...Let's take this equation from what I wrote above.
\[64=r^{n−1}\]
\[64=2^{n-1}\]
\[(n-1)=\frac{\log 64}{\log 2}\]

Answer 10

Plug that into your calculator and find what that equals to.

Answer 11

waht the RHS (Right Hand Side) equals to sorry.

Answer 12

what*

Answer 13

@XeiXei

Answer 14

i don't know how to solve it >3<

Answer 15

Mate, have you learnt anything on logarithms?

Answer 16

Because if you're learning series and sequences, you must of touched on the topic of logarithms.

Answer 17

I probably helped you on 90-95% of the working out on this question. The last 5% is probably the most easiest part of this question. When you're up to this stage in maths there's no sugar coating and saying this is difficult etc. If you're at this caliber of maths, you should know that it ain't going to get easier.

Answer 18

@XeiXei

Answer 19

Have you learnt he topic Logarithms.

Answer 20

?*

Answer 21

i think r = 191/127

Answer 22

No. That's incorrect. Both numbers are positive integers.

Answer 23

oppsss.. my mistake, r is not 191/127 :)
but, from ur information :
a*r^(n-1) = 192
3 * r^(n-1) = 192
r^(n-1) = 64
r^n/r = 64
r^n = 64 r .... (1)

Answer 24

Sn = 381
a * (r^n - 1)/(r-1) = 381
3 (r^n - 1)/(r-1) = 381
(r^n - 1)/(r-1) = 381/3
(r^n - 1)/(r-1) = 127 ... (2)

Answer 25

Ummm...that's gives you r=2....

Answer 26

that*

Answer 27

exactly the same thing. You just went an extra step to make it making r^n the subject instead of r^{n-1}

Answer 28

to making*