Tales by Dots and Lines — Question 1

(i) The mean of the first 50 natural numbers. The first 50 natural numbers. 1, 2, 3,………, 50 The sum of n natural numbers is \(\frac{n(n+1)}{2}\) So for n = 50 the sum = \(\frac{50(50+1)}{2}\) = \(\frac{50 \times 51}{2}\) = 1275 Mean = \(\frac{\text { Sum of the first } 50 \text { natural numbers }}{\text { Total number of natural numbers }}\) = \(\frac {1275}{50}\) = 25.5 Observations: The mean of the first n natural numbers is always \(\left(\frac{n+1}{2}\right)\) For n = 50, \(\frac {51}{2}\) = 25.5 (ii) Mean of the first 50 odd numbers. The first 50-odd numbers 1, 3, 5, 7, 9,…………, 99 Sum of the first n odd numbers = n 2 So for n (= 50) the sum = (50) 2 = 2500 Mean = \(\frac {2500}{50}\) = 50 Observations: The mean of the first n odd numbers is always n. For 50-odd numbers, the mean is 50. (iii) Mean of the first 50 multiples of 4. The first 50 multiples of 4 = 4, 8, 12, 16,………, 200 Sum of multiples of 4 = 4 + 8 + 12 + ……. + 200 = 4(1 + 2 + 3 + ….. + 50) = 4 × (1275) = 5100 Mean = \(\frac {5100}{50}\) = 102 Observations: The mean of the first n multiples of 4 is 4 times the mean of the first n natural numbers. Mean = 4 × 25.5 = 102