Задача к ЕГЭ на тему «Сложные задачи на числовые выражения» №2

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Найдите значение выражения $1 1 1 1 1 1 1 1 ---- + ---- + ---- + ---- + ---- + ----+ ----+ ------ 1 ⋅ 3 2 ⋅ 4 3 ⋅ 5 4 ⋅ 6 5 ⋅ 7 6 ⋅ 8 7 ⋅ 9 8 ⋅ 10$ . Ответ округлите до сотых.

--1- + --1- + -1--+ -1--+ -1--+ -1--+ -1-- + --1---= 1 ⋅ 3 (2 ⋅ 4 3 ⋅ 5 4 ⋅ 6 5 ⋅ 7 6 ⋅ 8 7 ⋅ 9 8 ⋅ 10 ) 1 2 2 2 2 2 2 2 2 = --⋅ ---- + ---- + ---- + ----+ ----+ ----+ ----+ ------ . 2 1 ⋅ 3 2 ⋅ 4 3 ⋅ 5 4 ⋅ 6 5 ⋅ 7 6 ⋅ 8 7 ⋅ 9 8 ⋅ 10

Так как для всякого $n ∈ ℕ$ верно

2 1 1 ---------- = -- − ------, n ⋅ (n + 2) n n + 2

то $1 ( 2 2 2 2 2 2 2 2 ) --⋅ ---- + ---- + ----+ ----+ ----+ ---- + ---- + ------ = 2 1 ⋅ 3 2 ⋅ 4 3 ⋅ 5 4 ⋅ 6 5 ⋅ 7 6 ⋅ 8 7 ⋅ 9 8 ⋅ 10$
$( ) 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1- 1-- = 2 ⋅ 1 − 3 + 2 − 4 + 3 − 5 + 4 − 6 + 5 − 7 + 6 − 8 + 7 − 9 + 8 − 10 =$
$1 ( 1 1 1 1 ) 4 = --⋅ --+ --− --− --- = --+ 0,2 = 0,(4) + 0,2 = 0,6(4) 2 1 2 9 10 9$ . После округления получим $0,64$ .