15. A tree seedling with extremely strong vitality, in the first week, grows 2 branches on the trunk (as shown in Figure 1), in the second week, it grows a new branch on each of the originally grown branches (as shown in Figure 2), in the third week, it grows a new branch on each of the branches that grew in the second week (as shown in Figure 3). The tree seedling grows according to this pattern, by the tenth week when new branches have grown, there will be $\qquad$ branches in total.
【Answer】2046 branches
【Key Point】Sum of a Geometric Sequence
【Analysis】
In the first week, the tree grows $1 \times 2$ new branches, totaling 2 branches;
In the second week, the tree grows $2 \times 2=2^{2}$ new branches, totaling $2+2^{2}$ branches;
In the third week, the tree grows $2^{2} \times 2=2^{3}$ new branches, totaling $2+2^{2}+2^{3}$ branches; and so on
In the tenth week, the tree grows $2^{10}$ new branches, totaling $2+2^{2}+2^{3} \cdots+2^{10}$ branches;
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Since $2+2^{2}+2^{3}+\cdots+2^{10}=2^{11}-2=2046$,
Therefore, after the new branches grow in the tenth week, there are a total of 2046 branches.