The Value of 100! to 110! Factorial Problem

  • Context: Undergrad 
  • Thread starter Thread starter Cosmos
  • Start date Start date
  • Tags Tags
    Factorial
Click For Summary

Discussion Overview

The discussion revolves around the evaluation of the expression 100! - 101! + 102! - 103! ... - 109! + 110!, exploring its value and the methods to simplify or calculate it. Participants engage in mathematical reasoning, attempting to manipulate the factorial terms and assess the size of the resulting number.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants suggest taking 100! as a common factor to simplify the expression, but express uncertainty about the subsequent steps.
  • One participant proposes rewriting the sum in a grouped form to facilitate calculation, indicating that the result will be large.
  • Another participant questions whether taking out 110! as a factor would be beneficial.
  • Several participants provide large numerical outputs for 110! and discuss the accuracy of their calculations, referencing Stirling's approximation and its precision.
  • There is mention of the number of digits in the result, with some questioning if all digits are necessary.
  • One participant expresses confusion about a specific large number presented in the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to evaluate the expression or the necessity of all digits in the result. Multiple competing views and approaches remain present throughout the discussion.

Contextual Notes

Some participants express uncertainty about the accuracy of their calculations and the implications of using Stirling's approximation, highlighting potential limitations in precision and assumptions about factorial growth.

Cosmos
Messages
18
Reaction score
1
What do you think is the value of
100!-101!+102!-103!...-109!+110!
:biggrin:
 
Mathematics news on Phys.org
n! = (n-1)!n. Take 100! as a common factor and go from there...
 
Come on man! i tried ... doesn't work...
 
Cosmos said:
Come on man! i tried ... doesn't work...
Show us that you tried...
 
when you take out 100! common out...then you are left with (1-101+102 times 101...) which afterwards...don't know man...addition is not the way out as it gives you a very humongous number...
 
Cosmos said:
What do you think is the value of
100!-101!+102!-103!...-109!+110!
:biggrin:
It's going to be pretty large.

Write the sum this way:

S = 110! - 109! + 108! - 107! + 106! - 105! + 104! - 103! + 102! - 101! + 100!

which can be grouped:

S = (110! - 109!) + (108! - 107!) + ... + (102! - 101!) + 100!

Now, take the term (110! - 109!) = (110 * 109! - 109!) = (110 - 1) * 109! = 109 * 109!

You can telescope the other terms in this sum in a similar fashion.

S = 109 * 109! + 107 * 107! + 105 * 105! + 103 * 103! + 101 * 101! + 100!

You can manipulate the terms in the sum above in a similar manner, but the result is clear:

S is a pretty big number no matter how you slice it.

Were you thinking that S would not be such a large number?
 
Does it help any if you take out 110! as a factor?
 
Are all 179 digits required?
 
17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000
 
  • Like
Likes   Reactions: pbuk
  • #10
micromass said:
17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000
Wow! And I was tempted to answer simply O(1). However, Stirling gave me 1.58...for 110! Would be interesting to know whether the calc.exe isn't precise enough or the margin in Stirling's formula is larger than I thought.
 
  • #11
fresh_42 said:
Wow! And I was tempted to answer simply O(1). However, Stirling gave me 1.58...for 110! Would be interesting to know whether the calc.exe isn't precise enough or the margin in Stirling's formula is larger than I thought.

According to the program I just wrote:
110!=15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000
So Stirling definitely is accurate here.
 
  • #12
micromass said:
17038855571963704692695290461249778228462303133623533009426911791940783815733361939707507950770908256181833575228292258746464777211982419630317448315535360000000000000000000000000
Not sure what this number is.
 
  • #13
Using Mathematica:
In[2]:= 110! - 109! + 108! - 107! + 106! - 105! + 104! - 103! + 102! - 101! + 100!

Out[2]= 15739381947081460468710896569033260448048487750802968746988405111340773775128510600810783940010370922688077274739713895911222137779156961431310006359162880000000000000000000000000
You can do it yourself using WolframAlpha.
 

Similar threads

Undergrad Questions regarding Kurepa's Conjecture
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Uncovering the Mystery of Calorie Constants
  • · Replies 1 ·
Replies
1
Views
1K
MHB Find the Value of $f(102)$ for $f(x)$ of Degree 100
  • · Replies 4 ·
Replies
4
Views
2K
MHB Find Remainder of $2^{100}-1$ Divided by 1000
  • · Replies 1 ·
Replies
1
Views
3K
MHB Find the Value of n in a Unique Factorial Problem with (n+1)!/(n-1)! = 56
  • · Replies 3 ·
Replies
3
Views
3K
Calculating Significant Figures for Regulation Soccer Field Area
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Can the Ionians continue counting indefinitely while following their four rules?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Which Allocation Maximizes Factory Profits?
  • · Replies 1 ·
Replies
1
Views
2K
Schools Making sense of college class numbers?
  • · Replies 5 ·
Replies
5
Views
2K
High School Is there a relationship between the harmonic mean and the standard deviation?
  • · Replies 5 ·
Replies
5
Views
2K