Algebra exercise

Question

If $a^4+a^3+a^2+a+1=0$ find the value of $a^{2000}+a^{2010}+1$

I got this problem in a book and tried to solve it.I multiplied with suitable powers of a and added and subtracted alternatively to get $a^{2010}+a^{2008}+a^{2006}+a^{2005}+a^{2004}+a^{2002}+a^{2000}=0$ but i can't figure what to do.I tried to replace $a^{2005}$ by multiplying the parent equation by $a^{2003}$ but nothing useful came.

Any help would be appreciated.Thanks in advance.

Does this answer your question? If $x^2+x+1 = 0$ then find the value of $x^{1999}+x^{2000}$? — Servaes, Nov 06 '20 at 16:03
Or: https://math.stackexchange.com/questions/2408233/if-a-is-a-root-of-x2-x-1-simplify-1-a-a2-dots-a2017?rq=1 — Servaes, Nov 06 '20 at 16:05
Better yet: https://math.stackexchange.com/questions/2474724/1-sum-of-two-factorials-in-two-ways-2-value-of-a2010a20101-given — Servaes, Nov 06 '20 at 16:07

score 4 · Answer 1 · answered Nov 06 '20 at 07:37

4

Hint

$$(a-1)(a^4+a^3+a^2+a+1)=?$$

Now $a^{2000}=(a^5)^{400}=?$

answered Nov 06 '20 at 07:37

lab bhattacharjee

274,582

Ooh i get it you found value of $a^5$ – Math enthusiast Nov 06 '20 at 07:39

score -1 · Answer 2 · answered Nov 06 '20 at 08:08

-1

I used @lab bhattacharjee's hint. $\frac{a^5-1}{a-1}=0$ $a\not= 1$ and $a_k=e^{\frac{2\pi ik}{5}}$ for $k=1,2,3,4$

$a^{2000}+a^{2010}+1=\left(e^{2\pi i}\right)^{400k}+\left(e^{2\pi i}\right)^{402k}+1=3$

answered Nov 06 '20 at 08:08

ratatuy

141

Algebra exercise

2 Answers2