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 Which is larger? (Posted on 2008-06-30)
Which is larger, where n is a natural number:
`     99n + 100n   or   101n ?`

 See The Solution Submitted by pcbouhid Rating: 4.0000 (4 votes)

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 re: Near Analytic Solution - the proof you asked for | Comment 4 of 7 |
(In reply to Near Analytic Solution by K Sengupta)

Nice work, KS. The proof you asked for:

Evaluating (101^n - 99^n)/100^n, we get

2* (n/100 + n(n-1)(n-2)/(3!*100^3) + ...

You already shown that this expression (by your means) exceeds 1 for n = 50 and for n = 49.

To show that the expression is less than 1 for n = 48:

A = 2*{48/100 + (48*47*46)/(3!*100^3) + (48*47*46*45*44)/(5!*100^5) + ...} is less than

2*[48/100 + (48^3)/(1*2*3)*100^3) + (48^5)/(1*2*3)(2*3)*100^5) + (48^7)/(1*2*3)(2*3)(2*3)*100^7 + ...]

where [x] is the greatest integer less than x.

= 2*[48/100 + 1/6*(48/100)^3 + 1/(6^2)*(48/100)^5 +...]

< 2* {(48/100) / (1 - 1/6*(48/100)^2}

= 9600/9616 < 1.

This problem was posed in a USSR Math Olympiad, in the lates 40īs, when there wasnīt electronic spreadsheet available, yet.

Edited on June 30, 2008, 1:41 pm
 Posted by pcbouhid on 2008-06-30 13:30:07

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