in game 2, your score started at 2 and doubled every time. two expressions below represent your score after…

in game 2, your score started at 2 and doubled every time. two expressions below represent your score after 12 hits. select both\n□ $\frac{2 + 2+cdots+2 + 2}{12 \text{ times}}$\n□ $\frac{2cdot2cdotcdotscdot2cdot2}{12 \text{ times}}$\n□ $2cdot12$\n□ $2^{12}$\ndone

in game 2, your score started at 2 and doubled every time. two expressions below represent your score after 12 hits. select both\n□ $\frac{2 + 2+cdots+2 + 2}{12 \text{ times}}$\n□ $\frac{2cdot2cdotcdotscdot2cdot2}{12 \text{ times}}$\n□ $2cdot12$\n□ $2^{12}$\ndone

Answer

Explanation:

Step1: Analyze the score - changing rule

The initial score is 2 and it doubles every time. After the first hit, the score is $2\times2 = 2^{2}$, after the second hit, the score is $2\times2\times2=2^{3}$, and in general, after $n$ hits, the score is $2^{n + 1}$. After 12 hits, the score is $2\times2\times\cdots\times2$ (13 times of 2 multiplied together), which can also be written as $2^{13}$. But if we consider the pattern of the expressions given, starting from the initial 2 and doubling 12 times, we can think of it as multiplying 2 by itself 12 times after the first - step 2. The expression for a quantity that starts at 2 and doubles 12 times is equivalent to multiplying 2 by itself 12 times, which is $\underbrace{2\times2\times\cdots\times2}_{12\text{ times}}$. In exponential form, this is $2^{12}$.

Step2: Evaluate each option

  • Option 1: $\underbrace{2 + 2+\cdots+2}_{12\text{ times}}$ represents the sum of 12 twos, which is $2\times12$, not the score when doubling 12 times.
  • Option 2: $\underbrace{2\times2\times\cdots\times2}_{12\text{ times}}$ is correct as it represents the score when starting from 2 and doubling 12 times.
  • Option 3: $2\times12$ is the sum of 12 twos, not the result of doubling 2 twelve times.
  • Option 4: $2^{12}$ is also correct as it is the exponential form of $\underbrace{2\times2\times\cdots\times2}_{12\text{ times}}$.

Answer:

B. $\underbrace{2\times2\times\cdots\times2}_{12\text{ times}}$, D. $2^{12}$