{"paper":{"id":"site-3074","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":10,"source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-3074/zad/3204","paper_id":"site-3074","number":"3204","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczby a, b, c tworzą w podanej kolejności ciąg geometryczny. Suma tych liczb jest równa 93. Te same liczby, w podanej kolejności są pierwszym, drugim i siódmym wyrazem ciągu arytmetycznego. Oblicz a, b, c.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-3204.svg","solution_image":"img/site/3074/rozw-3204.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/3205","paper_id":"site-3074","number":"3205","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Wyznacz wzór na n-ty wyraz ciągu arytmetycznego wiedząc, że suma pierwszych pięciu jego wyrazów jest równa 10, a wyrazy trzeci, piąty i trzynasty tworzą w podanej kolejności ciąg geometryczny.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-3205.svg","solution_image":"img/site/3074/rozw-3205.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/3306","paper_id":"site-3074","number":"3306","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczby 64, x, 4 są odpowiednio pierwszym, drugim i trzecim wyrazem malejącego ciągu geometrycznego. Oblicz piąty wyraz tego ciągu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-3306.svg","solution_image":"img/site/3074/rozw-3306.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/5333","paper_id":"site-3074","number":"5333","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Wszystkie wyrazy ciągu geometrycznego (a_n), określonego dla n≥1, są dodatnie. Wyrazy tego ciągu spełniają warunek 6a_1 - 5a_2 + a_3 = 0. Oblicz iloraz q tego ciągu należący do przedziału <2√2,3√2>.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-5333.svg","solution_image":"img/site/3074/rozw-5333.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/5498","paper_id":"site-3074","number":"5498","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Rosnący ciąg arytmetyczny (a_n) jest określony dla każdej liczby naturalnej n≥1. Suma pierwszych pięciu wyrazów tego ciągu jest równa 10. Wyrazy a_3, a_5, a_{13} tworzą - w podanej kolejności - ciąg geometryczny. Wyznacz wzór na n-ty wyraz ciągu arytmetycznego (a_n).","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-5498.svg","solution_image":"img/site/3074/rozw-5498.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/6725","paper_id":"site-3074","number":"6725","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Funkcja f jest określona wzorem f(x)=1/x dla każdej liczby rzeczywistej x≠0. Oblicz wartość m, dla której liczby f(m), f(1), f(2) są - odpowiednio - pierwszym, drugim i trzecim wyrazem ciągu geometrycznego.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-6725.svg","solution_image":"img/site/3074/rozw-6725.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/6890","paper_id":"site-3074","number":"6890","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Iloraz skończonego ciągu geometrycznego jest równy 1/3, trzeci wyraz tego ciągu jest równy 1/9, a suma wszystkich wyrazów to 364/243. Oblicz, z ilu wyrazów składa się ten ciąg.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-6890.svg","solution_image":"img/site/3074/rozw-6890.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/6891","paper_id":"site-3074","number":"6891","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczby x, y, z, których suma jest równa 114, tworzą w podanej kolejności ciąg geometryczny. Liczby te są również wyrazami pewnego ciągu arytmetycznego (a_n), gdzie n≥1, w którym x=a_1, y=a_4 i z=a_25. Oblicz liczby x, y, z.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-6891.svg","solution_image":"img/site/3074/rozw-6891.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/6892","paper_id":"site-3074","number":"6892","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Trzy liczby, których suma jest równa 24, tworzą ciąg arytmetyczny. Po zwiększeniu ich odpowiednio o 4, 10 i 40 będą w tej samej kolejności tworzyły ciąg geometryczny. Oblicz te trzy liczby tworzące ciąg arytmetyczny.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-6892.svg","solution_image":"img/site/3074/rozw-6892.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3074/zad/7260","paper_id":"site-3074","number":"7260","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Kwadrat K_1 ma bok długości a. Obok niego rysujemy kolejno kwadraty K_2, K_3, K_4, … takie, że kolejny kwadrat ma bok o połowę mniejszy od boku poprzedniego kwadratu (zobacz rysunek). [rysunek] Wyznacz pole kwadratu K_12.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3074/zad-7260.svg","solution_image":"img/site/3074/rozw-7260.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}