All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Infinitely Continued III (Posted on 2010-02-04) Difficulty: 3 of 5
Evaluate:

√(6+2√(7+3√(8+4√(9+5√(10+ ....)))))

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution | Comment 1 of 7

Based on the following table, as more and more terms are included, the value converges on 4:

To interpret the max coefficient in the table, consider, for max coef = 1, the value is just sqrt(6); for max coef = 2 the value is sqrt(6 + 2*sqrt(7)); for 3, it's sqrt(6 + 2*(sqrt(7 + 3*sqrt(8)))); etc. so for example just the figures shown in the puzzle itself, ignoring the + ... ellipsis, is the value for max coef = 5.

max coef           value
1             2.449489742783178
2             3.360283116365224
3             3.724280930782559
4             3.877744687922239
5             3.944740317401024
6             3.974678677681773
7             3.988278318778271
8             3.994530823306873
9             3.997431999017166
10            3.998787961199304
11            3.999425460591936
12            3.999726646219953
13            3.999869529354955
14            3.999937553613783
15            3.999970038367433
16            3.999985593152085
17            3.999993059053196
18            3.999996650107969
19            3.999998380676827
20            3.999999216092098
21            3.999999620010474
22            3.99999981558073
23            3.999999910395935
24            3.999999956418789
25            3.99999997878278
26            3.999999989661293
27            3.999999994957959
28            3.999999997539148
29            3.999999998798061
30            3.999999999412541
31            3.999999999712688
32            3.999999999859397
33            3.999999999931153
34            3.99999999996627
35            3.999999999983467
36            3.999999999991892
37            3.999999999996022
38            3.999999999998047
39            3.999999999999041
40            3.999999999999529
41            3.999999999999768
42            3.999999999999886
43            3.999999999999944
44            3.999999999999972
45            3.999999999999987
46            3.999999999999993
47            3.999999999999996
48            3.999999999999998
49            3.999999999999999
50            4
51            4
52            4
53            4
54            4
55            4

from

DEFDBL A-Z
FOR max = 1 TO 55
  v = max * SQR(max + 5)
  FOR c = max - 1 TO 1 STEP -1
    v = c * SQR(v + c + 5)
  NEXT c
  PRINT max, v
NEXT


  Posted by Charlie on 2010-02-04 13:48:05
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information