Which of the following sequences is/are in arithmetic progression?

0, \frac{1}{2}, 1, \frac{3}{2}, 2

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1, 4, 9, 16, 25

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- \sqrt{2}, 0, \sqrt{2}, 2 \sqrt{2}, 3 \sqrt{2}

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5, 10, 20, 35, 55

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Solution

The correct options are
A $0, \frac{1}{2}, 1, \frac{3}{2}, 2$
C $- \sqrt{2}, 0, \sqrt{2}, 2 \sqrt{2}, 3 \sqrt{2}$
$(a) W e h a v e a_{2} - a_{1} = (\frac{1}{2}) - 0 = \frac{1}{2} a_{3} - a_{2} = 1 - (\frac{1}{2}) = \frac{1}{2} a_{4} - a_{3} = (\frac{3}{2}) - 1 = \frac{1}{2} i . e ., a_{n} - a_{n - 1} i s t h e s a m e e v e r y w h e r e . S o, t h e g i v e n l i s t o f n u m b e r s f o r m a n A P .$
$(b) W e h a v e a_{2} - a_{1} = 4 - 1 = 3 a_{3} - a_{2} = 9 - 4 = 5 a_{4} - a_{3} = 16 - 9 = 7 i . e ., a_{n} - a_{n - 1} i s n o t t h e s a m e e v e r y w h e r e . S o, t h e g i v e n l i s t o f n u m b e r s d o n o t f o r m a n A P .$
$(c) W e h a v e a_{2} - a_{1} = 0 - (- \sqrt{2}) = \sqrt{2} a_{3} - a_{2} = \sqrt{2} - 0 = \sqrt{2} a_{4} - a_{3} = 2 \sqrt{2} - \sqrt{2} = \sqrt{2} i . e ., a_{n} - a_{n - 1} i s t h e s a m e e v e r y w h e r e . S o, t h e g i v e n l i s t o f n u m b e r s f o r m a n A P .$
$(d) W e h a v e a_{2} - a_{1} = 10 - 5 = 5 a_{3} - a_{2} = 20 - 10 = 10 a_{4} - a_{3} = 35 - 20 = 15 i . e ., a_{n} - a_{n - 1} i s n o t t h e s a m e e v e r y w h e r e . S o, t h e g i v e n l i s t o f n u m b e r s d o n o t f o r m a n A P .$

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