# HP Forums

Full Version: Musings on the HP-70
You're currently viewing a stripped down version of our content. View the full version with proper formatting.

Fibonacci Sequence

Initialisation

DSP 0
CLR
STO M
1

Loop

M+
x<>y

Result

0.
1.
1.
2.
3.
5.
8.
13.
21.
34.

Explanation

\begin{aligned} x_{0} &= 0 \\ x_{1} &= 1 \\ \\ x_{n+1} &= x_{n} + x_{n-1} \\ \end{aligned}

Python Program

Code:
a, b = 0, 1 for k in range(10):     print(a)     a, b = b, a + b

References

Viète's formula for $$\pi$$

Initialisation

DSP 9
0.5
STO K
CLR
STO M
2
ENTER
ENTER
ENTER

Loop

x<>y
M+
K
yx
STO M
÷
×

Result

2.000000000
2.828427125
3.061467459
3.121445152
3.136548491
3.140331157
3.141277251
3.141513801
3.141572940
3.141587725
3.141591422
3.141592346
3.141592577
3.141592634
3.141592649
3.141592652
3.141592653
3.141592654
3.141592654

Explanation

$$\pi = 2 \cdot \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{2 + \sqrt{2}}} \cdot \frac{2}{\sqrt{2 + \sqrt{2 + \sqrt{2}}}} \cdots$$

Python Program

Code:
from math import sqrt p, q = 2, 0 for k in range(20):     print(f"{p:>.9f}")     q = sqrt(2 + q)     p *= 2 / q

References

• Can you guess the result?
• Can you come up with other interesting recipes?
Here's another one:

Euler's number

Initialisation

DSP 9
-1
STO K
13
STO M
1
ENTER
ENTER
ENTER

Loop

K
M+
÷
+

Result

1.000000000
1.083333333
1.098484848
1.109848485
1.123316498
1.140414562
1.162916366
1.193819394
1.238763879
1.309690970
1.436563657
1.718281828
2.718281828

Explanation

\begin{aligned} e &= 1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots \\ &= 1 + \frac{1}{1}\left(1 + \frac{1}{2}\left(1 + \frac{1}{3}\left(1 + \cdots \right) \right) \right) \\ \end{aligned}

Python Program

Code:
s = 0 for k in range(13, 0, -1):     s = 1 + s / k     print(f"{s:>.9f}")

References

I hope you realise in time when you have to stop.
Natural Logarithm

Example

$$\log(1.2) = \log(1 + 0.2) \approx 0.182321557$$

Initialisation

DSP 9
-1
STO K
0.2
ENTER
ENTER
ENTER
11
STO M
÷

Loop

1
K
M+
÷
x<>y
-
×

Result

0.018181818
0.016363636
0.018949495
0.021210101
0.024329408
0.028467452
0.034306510
0.043138698
0.058038927
0.088392215
0.182321557

Explanation

\begin{aligned} \log(1+x) &= x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}} - \cdots \\ &= x \cdot \left(\frac{1}{1} - x \cdot \left(\frac{1}{2} - x \cdot \left(\frac{1}{3} - \cdots \right) \right) \right) \end{aligned}

Python Program

Code:
x = 0.2 s = 0 for k in range(11, 0, -1):     s = x * (1 / k - s)     print(f"{s:>.9f}")

References
Reference URL's
• HP Forums: https://www.hpmuseum.org/forum/index.php
• :