1. **Stating the problem:** We are given a geometric sequence starting with terms 5, 5/2, 10, 20, ... and asked to find the next term. 2. **Understanding geometric sequences:** A geometric sequence has a constant ratio $r$ between consecutive terms, i.e., $a_{n+1} = a_n \times r$. 3. **Find the common ratio:** - From 5 to 5/2: $\frac{5/2}{5} = \frac{5}{2} \times \frac{1}{5} = \frac{1}{2}$ - From 5/2 to 10: $\frac{10}{5/2} = 10 \times \frac{2}{5} = 4$ Since the ratio is not constant, check if the sequence is geometric or if the problem refers to a different pattern. The problem states it's geometric, so let's check the ratio between 10 and 20: - $\frac{20}{10} = 2$ The ratios are $\frac{1}{2}, 4, 2$, which are not equal, so the sequence as given is not geometric unless the problem means something else. 4. **Re-examining the problem:** The question is in Arabic: "الحد التالي في المتتالية الهندسية هو" meaning "the next term in the geometric sequence is". The options are 25, 30, 40, 80. 5. **Assuming the sequence is geometric starting from 5:** Let's check the ratio between 5 and 5/2: - $r = \frac{5/2}{5} = \frac{1}{2}$ Check if multiplying 5/2 by 2 gives 10: - $5/2 \times 2 = 5$, not 10. Check if multiplying 5/2 by 4 gives 10: - $5/2 \times 4 = 10$, yes. Check if multiplying 10 by 2 gives 20: - $10 \times 2 = 20$, yes. So the ratio changes: $\frac{1}{2}, 4, 2$ which is inconsistent. 6. **Alternative approach:** The problem might have a typo or the sequence is not geometric. However, the question asks why the answer is 40. 7. **Check if 40 fits as the next term:** - From 20 to 40: ratio is $\frac{40}{20} = 2$. If the ratio from 10 to 20 is 2, and from 20 to 40 is 2, then from 10 onwards the sequence is geometric with ratio 2. 8. **Conclusion:** The sequence from 10 onwards is geometric with ratio 2, so the next term after 20 is $20 \times 2 = 40$. **Final answer:** The next term is **40**.