Consider the sequence consisting of the odd natural numbers 1, 3, 5, 7, 9, 11, 13,...

  

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Consider the sequence consisting of the odd natural numbers 1, 3, 5, 7, 9, 11, 13, 15... to complete parts (a) through (c) below. (a) What is interesting about the ratio of the sum of the first k odds compared with the sum of the next k odds? For example, if k = 2, then the sum of the first two odds is 1+3 and the sum of the next two odds is 5+7, so the 1+3 4 1 ratio of sums is the ratio. = 5+7 12 = 3 Try other values of k and make a conjecture about the ratios. Try k = 4. Find (b) Repeat part (a), but this time form ratios of the sum of the first k odds to the sum of the next 2k odds. Perform this for k = 4. Find the ratio. 1+3+5+7 9+11+13+15+17+19+21+23 Consider the sequence consisting of the odd natural numbers 1, 3, 5, 7, 9, 11, 13, 15... to complete parts (a) through (c) below. (a) What is interesting about the ratio of the sum of the first k odds compared with the sum of the next k odds? For example, if k = 2, then the sum of the first two odds is 1+3 and the sum of the next two odds is 5+7, so the 1+3 4 1 ratio of sums is the ratio. = 5+7 12 = 3 Try other values of k and make a conjecture about the ratios. Try k = 4. Find (b) Repeat part (a), but this time form ratios of the sum of the first k odds to the sum of the next 2k odds. Perform this for k = 4. Find the ratio. 1+3+5+7 9+11+13+15+17+19+21+23