ad a) a_1 = 0 a_2 = 3 a_3 = 6 a_n = 999 r = a_2 - a_1 = 3 a_n = a_1 + (n - 1) \cdot r 999 = 0 + (n - 1) \cdot 3 3n - 3 = 999 3n = 1002 n = 334 S = {{n(a_1 + a_n)} \over 2} S = {{334 \cdot (0 + 999)} \over 2} S = 167 \cdot 999 S = 166833 ad b) a_1 = 0 a_2 = 1 a_3 = 2 a_n = 499 r = a_2 - a_1 = 1 a_n = a_1 + (n - 1) \cdot r 499 = 0 + (n - 1) \cdot 1 n - 1 = 499 n = 500 S_1 = {{n(a_1 + a_n)} \over 2} S_1 = {{500 \cdot (0 + 499)} \over 2} S_1 = 250 \cdot 499 S_1 = 124750 b_1 = 0 b_2 = 5 b_3 = 10 b_n = 495 r = b_2 - b_1 = 5 b_n = b_1 + (n - 1) \cdot r 495 = 0 + (n - 1) \cdot 5 5n - 5 = 495 5n = 500 n = 100 S_2 = {{n(b_1 + b_n)} \over 2} S_2 = {{100 \cdot (0 + 495)} \over 2} S_2 = 50 \cdot 495 S_2 = 24750 Wszystkich liczb niepodzielnych przez 5 i mniejszych od 500 jest S_1 - S_2 = 124750 - 24750 = 100000 ad c) a_1 = 2 a_2 = 9 a_3 = 16 a_n = 198 r = a_2 - a_1 = 7 a_n = a_1 + (n - 1) \cdot r 198 = 2 + (n - 1) \cdot 7 2 + 7n - 7 = 198 7n = 203 n = 29 S = {{n(a_1 + a_n)} \over 2} S = {{29 \cdot (2 + 198)} \over 2} S = {{29 \cdot 200} \over 2} S = 29 \cdot 100 S = 2900
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