1. The last term of a geometric sequence with a common ratio of 2 is 112, and the sum of all terms is 217. How many terms of this sequence must be added? Also, calculate the first term of this sequence.

(10 points)
1. After inserting the data into the general term of the geometric sequence, we get $112=a_{1} \cdot 2^{n-1}$. From the equation, we express the first term $a_{1}=\frac{112}{2^{n-1}}=\frac{224}{2^{n}}$. We insert the data into the formula for calculating the finite geometric series and get $217=\frac{a_{1}\left(2^{n}-1\right)}{2-1}$. We substitute the expressed first term $217=\frac{224}{2^{n}}\left(2^{n}-1\right)$. After rearranging the equation, we get $\frac{224}{2^{n}}=7$. We transform it into the equation $2^{n}=32$. We solve the exponential equation $2^{5}=32$ and get the solution of the equation $n=5$. We need to sum the first five terms of the geometric sequence. We calculate the first term $a_{1}=\frac{112}{2^{4}}=\frac{224}{2^{5}}$ and get the solution of the equation $a_{1}=7$.

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