Math 30 pure geometric series

[cathoderay]

helo this is a homework problem i got in math 30 pure
i got an answer but i would like to know how to get it by using a formula?

The exercise gose like this:
Initially, a pendulum swings through an arc of 2feet. On each successive swing,the length of the arc is 0.9 of the previous length.

a. Determine the length of the arc after 10 swings

i did this to find the answer:
using the formula Sn= a((r^n)-1)/r-1 were
Sn=?
a=2
r=0.9
n=10

Sn=2((0.9^10)-1)/0.9-1
Sn= 13.03 feet.

the aswer will be 13.03 feet

B. On which swing is the length of the arc first less than 1 foot?

I got that it was on the 7th swing were it is 0.9565938. but i got it by multipliying the 2feet by 0.9 and then its answer x 0.9 egain and so on ...until it give me an answer lower than 1...

how can i get this by using a formula(wich formula should i use) ??

C. After the 15 swings,Determine the total length that the pendulum will have swung.

using the same formula as a I can get this answer but rather than using n=10 it will be n=15
right??

thanks in advance for ur help.

 

[Shooting Star]

a. You have used the formula for the sum of a GP, where you should only use the formula to find the 11th term. The first term 'a' is given, and 'r' too.

b. Use the same formula for the nth term. Find n such that $$t_{n-1} > 1$$, but $$t_{n} < 1.$$

c. This is where you use the formula for the sum Sn.

 

[Dr Zoidburg]

A. Think about it for a second:
If each successive swing is less than the one prior, then how could the 10th swing be 11 feet more than the 1st?
Looks to me like you've got the total sum of the ten swings, not the length of the 10th.


B. We know that the 1st swing is 2 foot. Each successive swing is 0.9 less.
Therefore, the arc on the 5th swing, for eg, is 2*0.9^5 = 1.18 foot.
For this problem, you're given everything but the swing number.
ie.
2*0.9^x = 1 foot.

From there, it's a matter of using logs to solve. But remember to round the answer, as it does ask which swing!

 

[HallsofIvy]

The formula for the nth term of a geometric series is ar^n-1 where a is the first term and r is the common factor. In your case, a= 2, r= 0.9. That is, of course, just multiplying by the common factor the correct number of times as you did.

As has been pointed out, the first question asks for the length of the 10 the swing, not the total distance the pendulum has swung. That's just 2(0.9)⁹.
For (b) you have to solve 2(0.9)ⁿ< 1 which, as Dr. Zoidburg said, is just solving 2(0.9)ⁿ= 1 and rounding up.
(c) does as for the total distance swung so you have the right formula for that.