Ex 8.1, 1 - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

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Ex 8.1, 1 (i) Find the first five terms of the sequence in which the nth term is given by (i) 𝑡_𝑛=3𝑛−4 We need to find first five terms i.e. 𝒕_𝟏, 𝒕_𝟐, 𝒕_𝟑, 𝒕_𝟒, 𝒕_𝟓 Given 𝑡_𝑛=3𝑛−4 Putting n = 1 in 𝒕_𝒏 𝑡_1=3(1)−4 = 3 – 4 = –1 Putting n = 2 in 𝒕_𝒏 𝑡_2=3(2)−4 = 6 – 4 = 2 Putting n = 3 in 𝒕_𝒏 𝑡_3=3(3)−4 = 9 – 4 = 5 Putting n = 4 in 𝒕_𝒏 𝑡_4=3(4)−4 = 12 – 4 = 8 Putting n = 5 in 𝒕_𝒏 𝑡_5=3(5)−4 = 15 – 4 = 11 Thus, first 5 terms are –1, 2, 5, 8, 11 Ex 8.1, 1 (ii) Find the first five terms of the sequence in which the nth term is given by (ii) 𝑡_𝑛=2−5𝑛 We need to find first five terms i.e. 𝒕_𝟏, 𝒕_𝟐, 𝒕_𝟑, 𝒕_𝟒, 𝒕_𝟓 Given 𝑡_𝑛=2−5𝑛 Putting n = 1 in 𝒕_𝒏 𝑡_1=2−5(1) = 2 – 5 = –3 Putting n = 2 in 𝒕_𝒏 𝑡_2=2−5(2) = 2 – 10 = –8 Putting n = 3 in 𝒕_𝒏 𝑡_3=2−5(3) = 2 – 15 = –13 Putting n = 4 in 𝒕_𝒏 𝑡_4=2−5(4) = 2 – 20 = –18 Putting n = 5 in 𝒕_𝒏 𝑡_5=2−5(5) = 2 – 25 = –23 Thus, first 5 terms are –3, –8, –13, –18, –23 Ex 8.1, 1 (iii) Find the first five terms of the sequence in which the nth term is given by (iii) 𝑡_𝑛=𝑛^2−2𝑛+3 for 𝑛≥1. We need to find first five terms i.e. 𝒕_𝟏, 𝒕_𝟐, 𝒕_𝟑, 𝒕_𝟒, 𝒕_𝟓 Given 𝑡_𝑛=𝑛^2−2𝑛+3 Putting n = 1 in 𝒕_𝒏 𝒕_𝟏=1^2−2(1)+3 = 1 – 2 + 3 = 4 – 2 = 2 Putting n = 2 in 𝒕_𝒏 𝒕_𝟐=2^2−2(2)+3 = 4 – 4 + 3 = 0 + 3 = 3 Putting n = 3 in 𝒕_𝒏 𝒕_𝟑=3^2−2(3)+3 = 9 – 6 + 3 = 12 – 6 = 6 Putting n = 4 in 𝒕_𝒏 𝒕_𝟒=4^2−2(4)+3 = 16 – 8 + 3 = 19 – 8 = 11 Putting n = 5 in 𝒕_𝒏 𝒕_𝟓=5^2−2(5)+3 = 25 – 10 + 3 = 28 – 10 = 18 Thus, first 5 terms are 2, 3, 6, 11, 18