CPhill

I think they meant this, Melody......

3^(1/2) * 3^(5/2)  =    3^(1/2 + 5/2)  = 3^(6/2) = 3^(3)  = 27

1/2 + 1/3 +......+ 1/x  > 3/2

5/6 + 1/4 + 1/5 +.......+ 1/x > 3/2

1/4 + 1/5 +......+ 1/x > 2/3

9/20 + 1/6 + 1/7 > 2/3

1/6 + 1/7 > 13/60

1/7 > 1/20

So.....7 is the smallest integer

LOL!!!!!

(a) One possibility is 8.45

Rounded to the nearest integer = 8

Rounded to the nearest tenth = 8.5

And rounding this to the nearest integer = 9

(b) The possible numbers are 8.45 to 8.49, inclusively

.12........ = 12/99

.345...... = 345/999

So......   12/99 + 345/999  = [999(12) + 99(345)] / [99 *999] = 46143 / 98901 = .446557..........

So....the sum of the integers in the repeating part is  31

.666666.....   = 6/9  = 2/3

And  .606060.....  = 60/99

So

(2/3) (60/99)  =  (2/99)(60/3) = (2/99)(20) = 40/99 = .040404.........

So.......the sum of the first 15 digits  = 7(4) + 7(0) + 0 = 28

HAHAHA!!!....love that pic, Dbg.....!!!........{and the info on Ludwig van.....}

10r+r^2=120     rearrange

r^2 + 10r  - 120pi = 0     ........using the onsite solver, we have...

$${{\mathtt{r}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{10}}{\mathtt{\,\times\,}}{\mathtt{r}}{\mathtt{\,-\,}}{\mathtt{120}}{\mathtt{\,\times\,}}{\mathtt{\pi}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{r}} = {\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{24}}{\mathtt{\,\times\,}}{\mathtt{\pi}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}}}{\mathtt{\,-\,}}{\mathtt{5}}\\
{\mathtt{r}} = {\sqrt{{\mathtt{5}}}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{24}}{\mathtt{\,\times\,}}{\mathtt{\pi}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}}}{\mathtt{\,-\,}}{\mathtt{5}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{r}} = -{\mathtt{25.049\: \!716\: \!168\: \!334\: \!532}}\\
{\mathtt{r}} = {\mathtt{15.049\: \!716\: \!168\: \!334\: \!532}}\\
\end{array} \right\}$$

We can write this  infinite sum, thusly......

S = a₁ / [1 - d]    where a₁ is the first term and "d" is the common difference ...(1/5) in this case...so we have

(1/5) / [ 1 - 1/5)  =   (1/5) / (4/5)  =  (1/4)  = .25