1. ## Sequences and series

I have a very small understanding of this, please help me with these questions. Thank you so so much!

1. As a pendulum swings, air resistance and friction cause the extent of the swings to decrease. Suppose that the tip of the pendulum travels 120 cm on the first swing, 100 cm on its second, 83⅓ on its third and so on.
a) Find and state the appropriate mathematical model that enables the total distance travelled by tip of the pendulum after 10 swings to be found.
b) Calculate this total distance. State your answer to an appropriate accuracy.
c) Find the total distance travelled by the tip of the pendulum as it comes to rest.

2. Use the sum of the first five terms of the series for 1n (1 + x) to find an approximate value for 1n (1.1). Give your answer to 5 significant figures.

3. Use a series to find a value of (1.05)^5 correct to 4 decimal places.

2. Originally Posted by LilDragonfly
2. Use the sum of the first five terms of the series for 1n (1 + x) to find an approximate value for 1n (1.1). Give your answer to 5 significant figures.
Since, for $|x|<1$
$\ln|1+x|=x-\frac{x^2}{2}+\frac{x^3}{3}-...$
You need, to find $\ln (1.1)$ which means,
$(.1)-\frac{(.1)^2}{2}+\frac{(.1)^3}{3}-\frac{(.1)^4}{4}+\frac{(.1)^5}{5}$ these are first 5 terms.

3. Originally Posted by LilDragonfly
1. As a pendulum swings, air resistance and friction cause the extent of the swings to decrease. Suppose that the tip of the pendulum travels 120 cm on the first swing, 100 cm on its second, 83⅓ on its third and so on.
a) Find and state the appropriate mathematical model that enables the total distance travelled by tip of the pendulum after 10 swings to be found.
b) Calculate this total distance. State your answer to an appropriate accuracy.
c) Find the total distance travelled by the tip of the pendulum as it comes to rest.
This seems to me like a "geometric series". It starts,
120,100,83.3,...
Notice the common ratio is $.8\bar 3$ or $\frac{5}{6}$.
The the mathematical model is as said above namely that it follows the geometric series. Further, since the common ratio is $5/6$ and the first term is 120, we can state that on its $n$th swing the pendulum reaches $120(5/6)^{n-1}$. Check this out for the first three terms to see that it works.
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Next, to find the distance we need to sum the series.
Notice it would be an infinite series because our model (which is not completely perfect) never reaches zero. Thus,
$120(5/6)^0+120(5/6)^1+120(5/6)^2+...$
$120\left( (5/6)^0+(5/6)^1+(5/6)^2+...\right)$
Since, $|5/6|<1$ the series is convergent to,
$120\left(\frac{1}{1-5/6}\right)=720$

4. Thank you so much for your help so far