13.3 Homework: arithmetic and geometric series

In class we found a formula for the arithmetic series $3+5+7+9+11+13+15+17$ , by using a “trick” that works on all arithmetic series. Use that same trick to find the sum of the following series:

$10+13+16+19+22...100$

(Note that you will first have to figure out how many terms there are—that is, which term in this series 100 is. You can do that by using our previously discovered formula for the $n^{\mathrm{th}}$ term of an arithmetic sequence!)

Now I want you to take that same trick, and apply it to the general arithmetic series. Suppose we have an arithmetic series of $n$ terms, starting with $t_1$ and ending (of course) with $a_n$ . The common difference is $d$ , so the second term in the series is $t_1+d$ , and the third is $t_1+2d$ , and so on. So our whole series looks like…

$t_1+(t_1+d)+(t_1+2d)+\; \dots \; +(t_n-d)+t_n$

• A

  Find the general formula for the sum of this series

• B

  Use that general formula to add up $1+2+3+4+5$ . Did it come out right?

• C

  Use that general formula to add up all the even numbers between 1 and 100

In class we found a formula for the geometric series $2+6+18+54+162+486+1458$ , by using a completely different “trick” that works on geometric series. Use that same trick to find the sum of the following series:

$4+20+100+500+2500+...+39062500$

Now I want you to take that same trick, and apply it to the general geometric series. Suppose we have a geometric series of $n$ terms, with a common ratio of $r$ , starting with $t_1$ and ending (of course) with $t_n$ , which is $t_1{r}^{n-1}$ . The common ratio is $r$ , so the second term in the series is $t_{1}r$ , and the third term is $t_1{r}^2$ , and so on. So our whole series looks like…

$t_1+t_{1}r+t_1{r}^2+\dots +t_1{r}^{n-2}+t_1{r}^{n-1}$

• A

  Find the general formula for the sum of this series

• B

  Use that general formula to add up $1+2+4+8+16+32$ . Did it come out right?

• C

  Use that general formula to add up $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{\text{16}}+\frac{1}{\text{32}}$ . Did it come out right?

Suppose a ball is dropped from a height of 1 ft. It bounces back up. But each time it bounces, it reaches only $\frac{9}{\text{10}}$ of its previous height.

• A

  The ball falls. Then it bounces up and falls down again (second bounce). Then it bounces up and falls down again (third bounce). How high does it go after each of these bounces?

• B

  How high does it go after the 100th bounce?

• C

  How far does it travel before the fourth bounce? (*You don’t need any fancy math to do this part, just write out all the individual trips and add them up.)

• D

  How far does it travel before the 100th bounce?