Recursive function capability in a calculator can be advantageous and can eliminate the need for programming, particularly for those not well-versed in programming. Below are a few examples based on the Advanced Learning Calculator (Android only).
For an arithmetic sequence, where a(n),S(n)=n/2*(a(1)+a(n))a(n), since the S(n) formula is already known, no programming is required. However, this can be converted into a recursive formula and programmed, as shown below.
a(n)=n
S(1)=a(1)=1
S(2)=a(2)+S(1)
S(3)=a(3)+S(2)
...
S(n)=a(n)+S(n-1) (The current series value is equal to the current sequence value plus the prior series value.)
This can be programmed as followed with the app. (...) is comment.
S(1)=1 (initial value)
S(n)=n+S(n-1) (recursive formula)
S(10) (answer is 55, or from formula, (1+10)*10/2)
For a complicated sequence, a(n)=3n³+5n²+7n+8, it will be hard to figure out formula and calculate. Recursive programming can be handy.
a(n)=3n³+5n²+7n+8
S(1)=a(1)
S(n)=a(n)+S(n-1)
S(10) (answer = 11465)
Factorials can be converted into recursive formulas.
f(n)=n*(n-1)*(n02)...2*1 or
f(1)=1
f(2)=2*1=2*f(1)
f(3)=3*f(2)
---
f(n)=n*f(n-1) (Current value = n* prior value)
This can be programmed as below:
f(1)=1
f(n)=n*f(n-1)
f(5) (answer is 120)
The Fibonacci sequence is a classic example of a recursive sequence and can be calculated using this app.
f(1)=1 (initial value)
f(2)=1 (initial value)
f(n)=f(n-1)+f(n-2) (recursive formula)
f(20) (answer =6765)
f(50) (answer 12586269025)
For a comparison, below is Python code. Please note def, if, else, : , return, print are not required in the above app.
def f(n):
if n<2:
return 1
else:
return f(n-1)+f(n-2)
print(f(20))
print(f(50)) (memory overflow error)
Here's a Python code for generating the Fibonacci sequence to avoid memory overflow(ChatGPT generated). This is more complicated than the calculator app.
def fibonacci(n, memo={}):
if n in memo:
return memo[n]
if n <= 1:
return n
else:
memo[n] = fibonacci(n-1, memo) + fibonacci(n-2, memo)
return memo[n]
If you're interested, the same author has also developed a keyboard app with an embedded scientific calculator. This could be useful for generating reports with full scientific calculator capability.
Above app is updated to handle multiple variables recursion. Ackermann function is an example. Details of how it works.