## Problem Of The Week 43

**Find the number of real roots (x+1)^x=x^(x+1)**

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Written by Implex

October 4, 2008 at 2:10 am

Posted in Algebra, Problem of the week

Tagged with Algebra, Problem of the week

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No solution!But I couldn’t find any method do prove it..

Celebrating LifeOctober 4, 2008 at 3:34 am

try again !!

outtimedOctober 4, 2008 at 5:02 am

One solution

-1

tyroOctober 4, 2008 at 5:45 am

what is the method?

outtimedOctober 4, 2008 at 6:12 am

I took log to the base x of both sides and solved it for x.

tyroOctober 4, 2008 at 6:19 am

But then x can’t be -1. so it could be no solution.

tyroOctober 4, 2008 at 7:14 am

there is a solution try again !

outtimedOctober 4, 2008 at 7:23 am

I tried constructing a graph for the equation. Infinite real roots?

Totally confused with the question. @ Outtimed can you please explain the answer.

riyaOctober 4, 2008 at 10:40 am

The graph is a straight line parallel to an axis

tyroOctober 4, 2008 at 11:49 am

The value of x or if we think of it as values , will lie between 2 and 3 as for 2 or less than 2 one side of the equation becomes greater than the other and for 3 or greater than 3 , the other side of the equation becomes greater.

tyroOctober 4, 2008 at 12:46 pm

x comes to be the xth root of (1+1/x). Thus this forms an infinite series.

tyroOctober 4, 2008 at 2:24 pm

log(x+1)/(x+1)=logx/x

take f(y)=logy/y

plot the graph

now check if you can find a solution or not

hint: the graph does not rise or dip monotonously!

outtimedOctober 4, 2008 at 4:24 pm

basically we get

(1+1/n)^n =n

so basically we get one root between 2 and 3 and later on as limit of left hand side is e when n approaches infinity.. so just one root..

VarunOctober 4, 2008 at 6:05 pm

plotting the graph for log y/y between y=2 to 4 we get a curve which first increases then decreases, the value is almost same for 2.3 and 3.3, around 0.157

tyroOctober 5, 2008 at 3:53 am