Theory question 106
Progressions(a) Find the sum of the Arithmetic Progression (AP) 1, 3, 5, … 101.
(b) Out of the 95 travellers interviewed, 7 travelled by bus and train only, 3 by train and car only and 8 travelled by all three means of transport. The number, x, of travellers who travelled by bus only was equal to the number who travelled by bus and car only. If 47 people travelled by bus and 30 by train: (i) represent this information in a Venn diagram; (ii) calculate the I. value of x; II. number who travelled by at least two means.
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(a) a = 1, d = 2: 101 = 1 + 2(n − 1) → n = 51
S = n/2 (a + l) = 51/2 × 102 = 2601
(b)(i) Three overlapping circles B, T, C in a universal set of 95: B only = x, B∩C only = x, B∩T only = 7, T∩C only = 3, all three = 8.
(ii) I. Bus: x + x + 7 + 8 = 47 → x = 16
II. At least two means = 7 + 3 + 8 + 16 = 34
Fill the Venn diagram from the centre (all three) outwards.