Question

Calculați media aritmetică a numerelor a și b :

a= 1/2 + 1/3 + 1/4 + ... + 1/2016

b= 1 + 3/6 + 6/9 + 9/12 + ... + 6045/6048



Va rog repede dau coroana

Answer 1

$a = \dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2016}$

$\begin{aligned}b &= 1+\dfrac{3}{6}+\dfrac{6}{9}+\dfrac{9}{12}+...+\dfrac{6045}{6048}\\ &= 1+\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+...+\dfrac{2015}{2016}\end{aligned}$

$\begin{aligned}a+b = 1&+\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+...+\dfrac{2015}{2016}+\\&+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2016}\end{aligned}$

$\begin{aligned}a+b &= 1+\dfrac{2}{2}+\dfrac{3}{3}+\dfrac{4}{4}+...+\dfrac{2016}{2016}\\&=1+1+1+1+\underset{de\,\, 2016\, \,ori}{\underbrace{...}}+1\\ &=1\cdot 2016\\ &= 2016\end{aligned}$

$\Rightarrow \mathrm{m}_{\mathrm{a}}(a,b) =\dfrac{a+b}{2} = \dfrac{2016}{2} = \boxed{1008}$