Given-a-b-c-N-prove-that-1-a-1-2a-1-b-1-2b-1-c-1-2c-2-

Question Number 7782 by Chantria last updated on 15/Sep/16

G i v e n a, b, c \in N; p r o v e t h a t

\frac{1 + a}{1 + 2 a} + \frac{1 + b}{1 + 2 b} + \frac{1 + c}{1 + 2 c} ⩽ 2

Commented by sou1618 last updated on 15/Sep/16

L e t f (n) = \frac{1 + n}{1 + 2 n} (n ⩾ 1)

f^{'} (n) = \frac{1 \times (1 + 2 n) - (1 + n) \times 2}{{(1 + 2 n)}^{2}}

f^{'} (n) = \frac{- 1}{{(1 + 2 n)}^{2}} < 0

s o

f (n) ⩽ f (1) (n ⩾ 1)

f (1) = \frac{2}{3}

\frac{1 + a}{1 + 2 a} + \frac{1 + b}{1 + 2 b} + \frac{1 + c}{1 + 2 c} ⩽ 3 f (n) ⩽ 2

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