MHB Determining $a_7$ with Given Conditions

[anemone]

The positive integers $a_1,\,a_2,\,\cdots,a_7$ satisfy the conditions $a_6=144$, $a_{n+3}=a_{n+2}(a_{n+1}+a_n)$, where $n=1,\,2,\,3,\,4$.

Determine $a_7$.

 

[kaliprasad]

The positive integers $a_1,\,a_2,\,\cdots,a_7$ satisfy the conditions $a_6=144$, $a_{n+3}=a_{n+2}(a_{n+1}+a_n)$, where $n=1,\,2,\,3,\,4$.

Determine $a_7$.

Spoiler

144 * 24 or 3456

as

Spoiler

$a_4 = a_3 ( a_2 + a_1) $

$a_5 = a_4 ( a_3 + a_2) $
= $a_3 ( a_2 + a_1) ( a_3 + a_2)$

$a_6 = a_5(a_4+a_3) $
= $a_3 ( a_2 + a_1) ( a_3 + a_2)( a_3 ( a_2 + a_1) + a_3))$
= $a_3^2 ( a_2 + a_1) ( a_3 + a_2)( a_3 ( a_2 + a_1+1)$

it is product of 5 numbers of which 2 are same and 2 differ by 1

we have 144 = 2^2 * 3 * 3 * 4

so $a_3= 2$, $a_2 = 1 $ and $a_1 = 2$
using relation above we can compute $a_4= 6$, $a_5 = 18 $ and $a_6 =144$ and $a_7$ = 144* 24