Here are the sequences presented in the image, labeled accordingly:

Sequences List:

1. 41. $a_n = \left(\frac{n + 1}{2n}\right) \left(1 - \frac{1}{n}\right)$
2. 42. $a_n = \left(2 - \frac{1}{2n}\right) \left(3 + \frac{1}{2n}\right)$
3. 43. $a_n = \frac{(-1)^{n+1}}{2n - 1}$
4. 44. $a_n = \left(\frac{1}{2}\right)^n$
5. 45. $a_n = \frac{2n}{\sqrt{n + 1}}$
6. 46. $a_n = \frac{1}{(0.9)^n}$
7. 47. $a_n = \sin\left(\frac{\pi}{2} + \frac{1}{n}\right)$
8. 48. $a_n = n\pi \cos(n\pi)$
9. 49. $a_n = \frac{\sin n}{n}$
10. 50. $a_n = \frac{\sin^2 n}{2^n}$

Would you like a detailed analysis of any specific sequence, such as determining limits, convergence, or behavior as $n \to \infty$?

Related Questions:

1. How can you determine if these sequences converge or diverge?
2. What is the limit of $a_n$ for $n \to \infty$ in sequence 43?
3. Does sequence 44 represent a geometric progression? What is its behavior?
4. How can you simplify the expression for sequence 42?
5. What role does trigonometric behavior play in sequence 48?

Tip:

For sequences involving trigonometric functions, consider periodicity and bounds to determine convergence.