Welcome back — missed you this week. Step 1: Calculate the number of DNA copies after 10 cycles. In PCR, the number of DNA copies doubles with each cycle. The given series $2, 4, 8, 16, \dots$ indicates that after $n$ cycles, there will be $2^n$ copies of DNA. For 10 cycles, the number of copies is: $$ \text{Copies} = 2^{10} $$ $$ \text{Copies} = 1024 $$ The number of DNA copies synthesized at the end of 10 cycles is $\boxed{\text{1024 copies}}$. Step 2: Calculate the number of DNA copies after 30 minutes. First, determine how many cycles can be completed in 30 minutes. Each cycle takes 2 minutes. $$ \text{Number of cycles} = \frac{\text{Total time}}{\text{Time per cycle}} $$ $$ \text{Number of cycles} = \frac{30 \text{ minutes}}{2 \text{ minutes/cycle}} $$ $$ \text{Number of cycles} = 15 \text{ cycles} $$ Now, calculate the number of DNA copies after 15 cycles: $$ \text{Copies} = 2^{15} $$ $$ \text{Copies} = 32768 $$ The number of DNA copies synthesized at the end of 30 minutes is $\boxed{\text{32768 copies}}$.