### Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

### Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

# Giants

##### Stage: 5 Challenge Level:

Many people sent in the answers that 9 10 is bigger than 10 9 , that 99 100 is bigger than 100 99 and that 999 1000 is bigger than 1000 999 .

Steve said that this problem can also be done via inequalities, for example

My calculator could just work out that $99^{50} = 6.05\times 10^{99}$.

So, $99^{100}$ must equal $(6.05\times 10^{99})$ squared.

Let $X = 99^{100}$ and $Y= 100^{99}$

So, $X > (6 \times 10^{99})^2 = 36 \times 10^{198} > 3 \times 10 \times 10^{198} = 3\times 10^{199}$

As a straight power of 10 we know that $Y=10^{198}$.

So $X> 30 Y$.

Steve also said that an alternative simple solution works along these lines:

$\frac{999^{1000}}{1000^{999}} = \left(\frac{999}{1000}\right)^{999}\times 999 \approx 0.368 \times 999 \approx 367$

So the numerator is about 367 times bigger than the denominator.

Here is a splendid solution by Rachel from Stamford High School.

9 10 = 3 486 784 401

10 9 = 1 000 000 000

Hence 9 10 > 10 9

The answers to the next two questions are too big to fit on the calculator. We need another method.

100 to the power of 99 is 1 with 2*99=198 zeros.

Numbers Number of digits
99 squared = 9801 4
99 cubed = 970299 6
99 to power 4 =96059601 8
99 to power 5 =9509900499 10

Assuming that this pattern continues then 99 to the power 100 starts with 9 and has 200 digits, but 100 to the power 99 has only 199 digits, so 99 to the power 100 is bigger.

1000 to the power of 999 is 1 with 3*999=2998 zeros.

Numbers Number of digits
999 squared = 998001 6
999 cubed = 997002999 9

Again assuming that this pattern continues, then 999 to the power 1000 starts with 9 and has 3000 digits, but 1000 to the power 999 has only 2999 digits, so 999 to the power 1000 is bigger.

Here is another method that does not depend on the patterns in Rachel's tables continuing on and on for larger and larger numbers.

Although 99 100 and 100 99 are too big to evaluate using a calculator we can find:

99 1/99 = 1.047509406

100 1/100 = 1.047128548.

Hence

99 1/99 > 100 1/100

and so, raising each side of this inequality to the power 9900 gives

99 100 > 100 99 .

Using the same method to compare 999 1000 and 1000 999 :

999 1/999 = 1.006937623

1000 1/999 = 1.006931669.

Hence

999 1/999 > 1000 1/1000

and in the same way as before, raising each side of the inequality to the power 999000 gives:

999 1000 > 1000 999