csat-suneung 2013 Q30

csat-suneung · South-Korea · csat__math-humanities 4 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis
On the coordinate plane, for natural numbers $n$, consider the region $$\left\{ (x, y) \mid 2^x - n \leq y \leq \log_2(x + n) \right\}$$ Let $a_n$ be the number of points in this region satisfying the following conditions. (가) The $x$-coordinate and $y$-coordinate are equal. (나) Both the $x$-coordinate and $y$-coordinate are integers. For example, $a_1 = 2, a_2 = 4$. Find the value of $\sum_{n=1}^{30} a_n$. [4 points] 
On the coordinate plane, for natural numbers $n$, consider the region
$$\left\{ (x, y) \mid 2^x - n \leq y \leq \log_2(x + n) \right\}$$
Let $a_n$ be the number of points in this region satisfying the following conditions.\\
(가) The $x$-coordinate and $y$-coordinate are equal.\\
(나) Both the $x$-coordinate and $y$-coordinate are integers.\\
For example, $a_1 = 2, a_2 = 4$. Find the value of $\sum_{n=1}^{30} a_n$. [4 points]
Paper Questions
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