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Mudipookkal Vadiyal Song Free 28

Free download Mudipookkal Vadiyal entomane ninte chirippokkal മുടിപ്പൂക്കള്‍ വാടിയാലെന്തോമനേ.Q:

Arithmetic sequence proof: a collection of numbers with a certain property

Suppose $0\leq a_1a_n$ and $a_n>0$ then $a_{n+1}>1/2$

So we have $$\sum_{i=n+1}^{\infty}a_i\leq\sum_{i=1}^{\infty}a_i-a_n=a_1-a_n$$.
After this point I’m lost. I cannot see how to use this to prove that $a_{n+1}>1/2$.

A:

Since $a_n\geqslant 0$ for every $n\geqslant 1$ and
$$\sum_{i=1}^\infty a_i=\sum_{i=1}^\infty a_{i+n}=a_1+\sum_{i=1}^n a_i\leqslant a_1+\sum_{i=1}^\infty a_i$$
you necessarily have
$$\sum_{i=1}^n a_i\leqslant a_1+\sum_{i=1}^\infty a_i$$
for every $n\geqslant 1$. In particular, there exists $k\geqslant 1$ such that
$$\sum_{i=1}^

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