Question

упростить 2/(√5+√3) + 2/(√6+√4)+...+ 2/(√41+√39)

Answer 1

$frac{2}{sqrt5+sqrt3} + frac{2}{sqrt6+sqrt4} + frac{2}{sqrt7+sqrt5} +...+ frac{2}{sqrt{40}+sqrt{39}} + frac{2}{sqrt{41}+sqrt{39}} =\\= frac{2(sqrt5-sqrt3)}{5-3} + frac{2(sqrt6-sqrt4)}{6-4} + frac{2(sqrt7-sqrt5)}{7-5} +...+ frac{2(sqrt{40}-sqrt{48})}{40-38} + frac{2(sqrt{41}-sqrt{39})}{41-39}=\\=sqrt5-sqrt3+sqrt6-sqrt4+sqrt7-sqrt5+sqrt8-sqrt6+sqrt9-sqrt7+$

$+sqrt{10}-sqrt8+...+sqrt{38}-sqrt{36}+sqrt{39}-sqrt{37}+sqrt{40}-sqrt{38}+$

 $+sqrt{41}-sqrt{39}=$ $sqrt{41}+sqrt{40}-sqrt{4}-sqrt3$