Let a_1, a_2, a_3… ,a_10 be in G.P.with a_i>0 for i=1, 2, … , 10 and S be the set of pairs (r, k), r, k∈ N (the set of natural numbers) for which | …

Let ${a}_{1}, {a}_{2}, {a}_{3}\ldots ,{a}_{10}$a1​,a2​,a3​…,a10​ be in $G.P.$G.P.with ${a}_{i}>0$ai​>0 for $i=1, 2, \ldots , 10$i=1,2,…,10 and $S$S be the set of pairs $(r, k), r, k\in N$(r,k),r,k∈N (the set of natural numbers) for which

$$|\begin{matrix}{\mathrm{log}}_{e}⁡{a}_{1}^{r} {a}_{2}^{k} & {\mathrm{log}}_{e}⁡{a}_{2}^{r}{a}_{3}^{k} & {\mathrm{log}}_{e}⁡{a}_{3}^{r}{a}_{4}^{k} \\ {\mathrm{log}}_{e}⁡{a}_{4}^{r} {a}_{5}^{k} & {\mathrm{log}}_{e}⁡{a}_{5}^{r}{a}_{6}^{k} & {\mathrm{log}}_{e}⁡{a}_{6}^{r}{a}_{7}^{k} \\ {\mathrm{log}}_{e}⁡{a}_{7}^{r}{a}_{8}^{k} & {\mathrm{log}}_{e}⁡{a}_{8}^{r}{a}_{9}^{k} & {\mathrm{log}}_{e}⁡{a}_{9}^{r}{a}_{10}^{k}\end{matrix}|=0$$∣loge​⁡a1r​a2k​loge​⁡a4r​a5k​loge​⁡a7r​a8k​​loge​⁡a2r​a3k​loge​⁡a5r​a6k​loge​⁡a8r​a9k​​loge​⁡a3r​a4k​loge​⁡a6r​a7k​loge​⁡a9r​a10k​​∣=0

Then the number of elements in $S,$S, is: