Let
![a_1,\ldots,a_8](/media/m/9/7/9/97963b861a419eddce03aa2c5b64743e.png)
be reals, not all equal to zero. Let
![\displaystyle c_n = \sum^8_{k=1} a^n_k](/media/m/3/e/3/3e3d09c9413ec40b4a1193eab9f7541a.png)
for
![n=1,2,3,\ldots](/media/m/6/8/5/685e606480c30a42cad948d06073cee7.png)
. Given that among the numbers of the sequence
![(c_n)](/media/m/9/c/c/9cc90d3e330e131e7fb3a01b0c28ae4f.png)
, there are infinitely many equal to zero, determine all the values of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
for which
![c_n = 0](/media/m/1/2/3/123fd6ecb4e8ad99f3f13c5e5f7de898.png)
.
%V0
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let $$\displaystyle c_n = \sum^8_{k=1} a^n_k$$ for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0$.