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 Hari Shankar's messages in the community 1 2 3 4 ... 109 110 111 GO Go to Page...
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 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

 Discussion Forums -> This Post 7 points    (1    in 2 votes )   [?]

Rewrite the problem by substituting 1 by (a+b+c) as having to prove that

Setting  and dividing by each factor on both sides by 2, we can further rewrite as

given x+y+z=1, to prove that  which is well known and easy as

Multiplying the above three inequalities concludes the solution

 Discussion Forums -> This Post 12 points    (2    in 3 votes )   [?]

 Discussion Forums -> This Post 7 points    (1    in 2 votes )   [?]

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

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 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

quite obviously (0,1).

If b and c are equal, you can make a as small as you please (as we only need a>b-c)

If a is very large then . But again from triangle inequality a<b+c. Hence the range is (0,1)

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Congratulations Vichar!

Well as you have very little time left, I would suggest visiting mathlinks.ro

This is an international forum is most used by students preparing for math olympiads across the world

There is a large bank of questions from contests across the world over several years which you will find very useful

@rishabh.

The angles A,B,C of the triangle are

Hence

So the questions wants us to prove that

But this is true as

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Let the number of terms be n. Then the sum of terms of the AP is

Suppose the common ratio of the terms is r, then

Now the sum of terms of the GP can be written as

The typical bracketed term is

We will now prove that this is less than or equal to a+b =

WLOG we may assume that r>1 (otherwise we read the sequence backwards)

So we are to prove that

Hence, each bracketed term is less than or equal to a+b.

Hence the entire sum is less than or equal to  thus proving the statement

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Since when did we abandon the convention that the remainder is to be less than the divisor?

 Discussion Forums -> This Post 10 points    (2    in 2 votes )   [?]

Hence the given expression equals . Hence the remainder is 1

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

The solutions above are incorrect

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

first of all, we are talking sequences really and not functions, and you dont apply derivatives in the conventional sense to them.

Again f(n) being finite has nothing to do with it. Consider f(n) = sin n as n goes to infinity. The sequence is bounded, but it does not converge to any specific value. So saying than sin n = sin (n+1) because as n becomes large n and (n+1) are almost equal would be an error. Clearly if your sequence goes 1,-1,1,-1,... you wouldnt say f(n) = f(n+1)!

That is why proving convergence is important. After that you are justfied in applying this as you are actually applying limits to both sides

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

That is not the right procedure.

Suppose I give you f(n+1) = f(n)+1, using this procedure you get 1=0. (!!!)

You first need to prove that the sequence is bounded. For this we note that for n>1, we have

Hence the sequence is bounded from below. Note that if a= 3, then all members of the sequence equal 3, otherwise they are all greater. We will now assume that

Also, we have

That means the sequence is monotonically decreasing. This sequence is known to converge (in fact to its least upper bound)

Now you can use  as done above to obtain the limit as 3

This method was quite popular among the Babylonians to compute the square root of any positive number

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

In fact, we can even give a closed form expression for the general term.

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

Its easy to see that

 Discussion Forums -> This Post 2 points    (0    in 1 votes )   [?]

The sum of 5 odd numbers is also odd. So that cannot be 50,. Pls chk the question

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]
 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

even shorter, ab+bc+ca =23 and abc=-15.

Hence

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