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# A mysterious series

Let $$a_1,a_2,...a_{100}$$ be real numbers each less than $$1$$,which satisfy, $a_1+a_2+.....a_{100} > 1$

$$1.$$ Let $$n_0$$ be the smallest integer $$n$$ such that $a_1+a_2+.....a_n > 1$ Show that the sums $$a_{n_0},a_{n_0}+a_{n_0-1} ,......,a_{n_0}+.....+a_1$$ are positive

$$2.$$Show that there exists two integers $$p$$ and $$q$$,$$p < q$$,such that the numbers $a_q,a_q+a_{q-1},....,a_q+.....+a_p$ and $a_p,a_p+a_{p+1},....,a_p+.....+a_q$

are all positive

3 years, 9 months ago

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For 1., we prove by contradiction. Suppose one the sums, WLOG $$(a_{n_0} + \dots + a_{n_0 - x})$$ is not positive., i.e.$$(\leq 0)$$

Now, As, $$a_1 + a_2 + \dots + a_{n_0} > 1$$

$$\Rightarrow (a_1 + a_2 + \dots + a_{n_0 - (x+1)}) + ( a_{n_0 - x} + \dots + a_{n_0}) > 1$$

$$\Rightarrow (a_1 + a_2 + \dots + a_{n_0 - (x+1)}) > 1 - ( a_{n_0 - x} + \dots + a_{n_0})$$

$$\Rightarrow (a_1 + a_2 + \dots + a_{n_0 - (x+1)}) > 1$$ (Since $$( a_{n_0 - x} + \dots + a_{n_0})$$ is not positive)

But this contradicts the fact $$a_{n_0}$$ is the smallest n for which $$a_1 + a_2 + \dots + a_n > 1$$.

Therefore our supposition is wrong and no sum is not positive, i.e. all the sums are positive

- 3 years, 9 months ago