Small challenge

[C.Ret]

Ah! That's nice, I really get a near close !
I try different ways to compute the 10th exponent but completely miss the two stage one.

With a little effort, the HP-28C/S can print the exact sequence without omission of any trailing zeros:

« FOR n  n  5  ^  SQ  PR1  NEXT »  'PRx¹º' STO  
« CR  STD  5 15 PRx¹º  12 SCI  16 24 PRx¹º   CR »
EVAL

(04-23-2023 12:58 AM)Valentin Albillo Wrote:  The ending digits (5,6,9,4,1,0,1,4,9,6,5,) which are the ending digits of consecutive squares, plus the ten zeros in 10000000000 are hinting enough at the correct expression.

I'm not as clever as Valentin. I did not know by hearth the list of the last digits of any square. I only used this trick to detect out certain multiples (of 2, 3 or 5 in most cases). But it's a good trick to use.

I found the general phrase while chatting with a new cyber friend of mine lining in San Francisco CA that has been very popular lately.

ChatGPT 3.5 (23 March) (c) OpenAI, L.L.C. Wrote:Based on the consecutive terms you provided, it appears that each term is an exact power of a fixed integer. More specifically, each term is an exact power of 10.

To see this, notice that the first term is 5^10, the second term is 6^10, the third term is 7^10, and so on. Therefore, the general term of the sequence can be expressed as u(n) = (n+4)^10.

Here is a table that shows the first 11 terms of the sequence, along with the formula u(n) = (n+4)^10 and the exact power of ten that each term represents:

Index (n)   Known Value    Formula: (n+4)^10   Power of Ten
k         9765625         5^10         10
k+1       60466176         6^10         10
k+2       282475249         7^10         10
k+3      1073741824         8^10         10
k+4      3486784401         9^10         10
k+5      1000000000         10^10         10
k+6     25937424601         11^10         10
k+7     61917364224        12^10         10
k+8    137858491849         13^10         10
k+9    289254654976        14^10          10
k+10   576650390625         15^10         10
As you can see, each known value is an exact power of 2 and 5, and the formula (n+4)^10 matches the known values perfectly.