What is the value of it? Thank you in advance.

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What is the value of it? Thank you in advance.

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Answer 1 · 2018-01-06T19:22:16

$a_{2014} = \frac{4}{5}$ $a_2014=4/5$

Explanation:

We know that:
$a_{1} = 2$ $a_1=2$
$a_{2} = 5$ $a_2=5$
$a_{n + 2} = \frac{1 + a_{n + 1}}{a_{n}}$ $a_(n+2)=(1+a_(n+1))/a_n$

Using this, we can find:
$a_{3} = \frac{1 + 5}{2} = \frac{6}{2} = 3$ $a_3=(1+5)/2=6/2=3$

$a_{4} = \frac{1 + 3}{5} = \frac{4}{5}$ $a_4=(1+3)/5=4/5$

$a_{5} = \frac{1 + \frac{4}{5}}{3} = \frac{\frac{5}{5} + \frac{4}{5}}{3} = \frac{\frac{9}{5}}{3} = \frac{9}{15} = \frac{3}{5}$ $a_5=(1+4/5)/3=(5/5+4/5)/3=(9/5)/3=9/15=3/5$

$a_{6} = \frac{1 + \frac{3}{5}}{\frac{4}{5}} = \frac{\frac{5}{5} + \frac{3}{5}}{\frac{4}{5}} = \frac{\frac{8}{5}}{\frac{4}{5}} = \frac{8}{5} \cdot \frac{5}{4} = \frac{8}{4} = 2$ $a_6=(1+3/5)/(4/5)=(5/5+3/5)/(4/5)=(8/5)/(4/5)=8/5*5/4=8/4=2$

$a_{7} = \frac{1 + 2}{\frac{3}{5}} = \frac{3}{\frac{3}{5}} = 3 \div \frac{3}{5} = 3 \cdot \frac{5}{3} = 5$ $a_7=(1+2)/(3/5)=3/(3/5)=3-:3/5=3*5/3=5$

The sequence has a loop of 5 numbers, $((2, 5,3,4/5,3/5),(a_1,a_2,a_3,a_4,a_5))$

Since 2015 is a multiple of 5, $a_2015=3/5$

So, $a_2014=4/5$

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