Question

Let us consider a sequence
`\rm a_(n)` as follows.


`\rm a_(n+1)=a_(n)(a^(2)_(n)-3a_(n)+3),\ a_(1)=10`
`\;`
Find
`a_(5).`​

Answer 1

Use the given recursive rule to find the second term, a₂ :

`a_2 = a_1 ({a_1}^2 - 3a_1 + 3) = 10 (10^2-30+3) = 730`

Do the same to find the next few terms:

`a_3 = a_2 ({a_2}^2 - 3a_2 + 3) = 730 (730^2 - 2190 + 3) = 387\,420\,490`

`a_4 = a_3 ({a_3}^2 - 3a_3 + 3) = 387\,420\,490 (387\,420\,490^2 - 1\,162\,261\,470 + 3)\\ = 58\,149\,737\,003\,040\,059\,690\,390\,170`

`a_5 = a_4 ({a_4}^2 - 3a_4 + 4) = \cdots \\ = 196\,627\\{}\,\,\,\,\,\,050\,475\,552\,913\,618\,075\,908\,526\\{}\,\,\,\,\,\,912\,116\,283\,103\,450\,944\,214\,766\\{}\,\,\,\,\,\,927\,315\,415\,537\,966\,391\,196\,810 \\ \approx 1.96 * 10^(27)`