A

1003085006

Level: 
A
A sequence \( \left(a_n\right)_{n=1}^{\infty} \) is defined by the recursive formula \( a_1=1\text{, }a_2=2;\ a_{n+2} = \frac12\left(a_{n+1}+a_n\right)\text{, }n\in\mathbb{N} \). What are its first five terms?
\( 1 \), \( 2 \), \( \frac32 \), \( \frac74 \), \( \frac{13}8 \)
\( 1 \), \( 2 \), \( \frac32 \), \( \frac47 \), \( \frac8{13} \)
\( 1 \), \( 2 \), \( 3 \), \( 7 \), \( 13 \)
\( 1 \), \( 2 \), \( \frac23 \), \( \frac47 \), \( \frac{13}8 \)

1003085005

Level: 
A
A sequence \( \left( a_n \right)_{n=1}^{\infty} \) is defined by the recursive formula \( a_1=1;\ a_{n+1}=\frac1{1+a_n}\text{, }n\in\mathbb{N} \). What are its first five terms?
\( 1 \), \( \frac12 \), \( \frac23 \), \( \frac35 \), \( \frac58 \)
\( 1 \), \( \frac12 \), \( \frac23 \), \( \frac34 \), \( \frac58 \)
\( 1 \), \( 2 \), \( \frac32 \), \( \frac53 \), \( \frac85 \)
\( 1 \), \( \frac12 \), \( \frac32 \), \( \frac35 \), \( \frac85 \)

1003085004

Level: 
A
A sequence \( \left(a_n\right)_{n=1}^{\infty} \) is defined by the recursive formula \( a_1=1;\ a_{n+1} = 3a_n\text{, }n\in\mathbb{N} \). What are its first five terms?
\( 1 \), \( 3 \), \( 9 \), \( 27 \), \( 81 \)
\( 3 \), \( 9 \), \( 27 \), \( 81 \), \( 243 \)
\( 1 \), \( 3 \), \( 6 \), \( 12 \), \( 24 \)
\( 1 \), \( 3 \), \( 9 \), \( 30 \), \( 90 \)

1003085003

Level: 
A
We were given the sequence \( \left(\sin\left(n\cdot\frac{\pi}2\right)\right)_{n=1}^{\infty} \). What are the first five terms?
\( 1 \), \( 0 \), \( -1 \), \( 0 \), \( 1 \)
\( 1 \), \( 0 \), \( 1 \), \( 0 \), \( 1 \)
\( -1 \), \( 0 \), \( 1 \), \( 0 \), \( 1 \)
\( 0 \), \( -1 \), \( 0 \), \( 1 \), \( 0 \)

1003085002

Level: 
A
We were given the sequence \( \left(\frac{n+3}{2n}\right)_{n=1}^{\infty} \). What are its first five terms?
\( 2 \), \( \frac54 \), \( 1 \), \( \frac78 \), \( \frac45 \)
\( \frac45 \), \( \frac78 \), \( 1 \), \( \frac54 \), \( 2 \)
\( 2 \), \( \frac45 \), \( 1 \), \( \frac87 \), \( \frac54 \)
\( \frac12 \), \( \frac23 \), \( \frac34 \), \( \frac45 \), \( \frac56 \)

1003085001

Level: 
A
We were given the sequence \( \left(\frac1{3^n}\right)_{n=1}^{\infty} \). What are its first five terms?
\( \frac13 \), \( \frac19 \), \( \frac1{27} \), \( \frac1{81} \), \( \frac1{243} \)
\( 3 \), \( 9 \), \( 27 \), \( 81 \), \( 243 \)
\( 3 \), \( 6 \), \( 9 \), \( 12 \), \( 15 \)
\( \frac13 \), \( \frac16 \), \( \frac19 \), \( \frac1{12} \), \( \frac1{15} \)

1003123503

Level: 
A
It is known that in the given time period one up to five pieces of the same product were sold in each of the 100 monitored shops. One piece was sold in \( 26 \) shops, \( 2 \) pieces in \( 64 \) shops, \( 3 \) pieces in \( 7 \) shops, \( 4 \) pieces in \( 2 \) shops and \( 5 \) pieces in one shop. How many pieces of the product were sold most frequently in the given shops? Choose the correct characteristic and its value.
Mode: \( 2 \) pieces
Arithmetic mean: \( 3 \) pieces
Median: \( 2 \) pieces
Median: \( 3 \) pieces
(Weighted) arithmetic mean: \( 1.88 \) pieces

1003123502

Level: 
A
At a weather station in Las Vegas temperature was measured every day at \( 7 \)pm during one month. The results are specified in the following table. \[ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text{Day} & 1. & 2. & 3. & 4. & 5. & 6. & 7. & 8. \\\hline \text{Temperature }(^{\circ}\mathrm{C}) & 24 & 22 & 21 & 26 & 22 & 23 & 21 & 23 \\\hline \\\hline \text{Day} & 9. & 10. & 11. & 12. & 13. & 14. & 15. & \\\hline \text{Temperature } (^{\circ}\mathrm{C}) & 21 & 26 & 20 & 23 & 24 & 19 & 21 & \\\hline \\\hline \text{Day} & 16. & 17. & 18. & 19. & 20. & 21. & 22. & 23. \\\hline \text{Temperature } (^{\circ}\mathrm{C}) & 21 & 20 & 26 & 23 & 24 & 22 & 23 & 26 \\\hline \\\hline \text{Day} & 24. & 25. & 26. & 27. & 28. & 29. & 30. & \\\hline \text{Temperature } (^{\circ}\mathrm{C}) & 25 & 23 & 22 & 25 & 27 & 26 & 22 & \\\hline \end{array} \] Determine the mode of the recorded temperatures.
\( 23 \,^{\circ}\mathrm{C} \)
\( 22\,^{\circ}\mathrm{C} \)
\( 21\,^{\circ}\mathrm{C} \)
\( 26\,^{\circ}\mathrm{C} \)
\( 21\,^{\circ}\mathrm{C} \) also \( 22\,^{\circ}\mathrm{C}\) also \(26\,^{\circ}\mathrm{C}\)

1003123501

Level: 
A
At a weather station in Las Vegas wind speeds were measured every day at \( 7 \)pm during one month. The results are specified in the following table. \[ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text{Day} & 1. & 2. & 3. & 4. & 5. & 6. & 7. & 8. \\\hline \text{Wind} (\mathrm{mps}) & 3 & 2 & 1 & 1 & 2 & 2 & 1 & 3 \\\hline \\\hline \text{Day} & 9. & 10. & 11. & 12. & 13. & 14. & 15. & \\\hline \text{Wind} (\mathrm{mps}) & 2 & 1 & 2 & 2 & 4 & 2 & 4 & \\\hline \\\hline \text{Day} & 16. & 17. & 18. & 19. & 20. & 21. & 22. & 23. \\\hline \text{Wind} (\mathrm{mps}) & 4 & 2 & 2 & 3 & 2 & 12 & 13 & 6 \\\hline \\\hline \text{Day} & 24. & 25. & 26. & 27. & 28. & 29. & 30. & \\\hline \text{Wind} (\mathrm{mps}) & 5 & 7 & 2 & 3 & 8 & 9 & 12 & \\\hline\end{array} \] Determine the median of the recorded wind speeds.
\( 2{.}5\,\mathrm{m/s} \)
\( 2\,\mathrm{m/s} \)
\( 4\,\mathrm{m/s} \)
\( 6\,\mathrm{m/s} \)