- #1

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What do you think is the value of

100!-101!+102!-103!.........-109!+110!

100!-101!+102!-103!.........-109!+110!

- Thread starter Cosmos
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- #1

- 18

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What do you think is the value of

100!-101!+102!-103!.........-109!+110!

100!-101!+102!-103!.........-109!+110!

- #2

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n! = (n-1)!n. Take 100! as a common factor and go from there...

- #3

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Come on man!!! i tried ... doesn't work....

- #4

Mark44

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Show us that you tried...Come on man!!! i tried ... doesn't work....

- #5

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- #6

SteamKing

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It's going to be pretty large.What do you think is the value of

100!-101!+102!-103!.........-109!+110!

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?

- #7

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Does it help any if you take out 110! as a factor?

- #8

fresh_42

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Are all 179 digits required?

- #9

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- #10

fresh_42

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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

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According to the program I just wrote: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.

110!=15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000

So Stirling definitely is accurate here.

- #12

SteamKing

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Not sure what this number is.

- #13

DrClaude

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Out[2]= 15739381947081460468710896569033260448048487750802968746988405111340773775128510600810783940010370922688077274739713895911222137779156961431310006359162880000000000000000000000000

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