Example 3 - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

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Example 3 Find the first four terms of the sequence given by the recursive rule u_1=1, 𝑢_𝑛=2𝑢_(𝑛−1)+3 for 𝑛≥2. Is 133 a term of this sequence? First, let’s find first four terms i.e. 𝒖_𝟏, 𝒖_𝟐, 𝒖_𝟑, 𝒖_𝟒 Given First term = 𝒖_𝟏 = 1 Second term = 𝑢_2 = 2𝑢_(2−1)+3 = 𝟐𝒖_𝟏+𝟑 = 2 × 1 + 3 = 2 + 3 = 5 Third term = 𝑢_3 = 2𝑢_(3−1)+3 = 𝟐𝒖_𝟐+𝟑 = 2 × 5 + 3 = 10 + 3 = 13 Fourth term = 𝑢_4 = 2𝑢_(4−1)+3 = 𝟐𝒖_𝟑+𝟑 = 2 × 13 + 3 = 26 + 3 = 29 Thus, first four terms of the sequence are 1, 5, 13, 29 Now, we are asked Is 133 a term of this sequence Since we are given a recursive rule, we have to find the next several terms to see if 133 is a term or now Let’s do that Fifth term = 𝑢_5 = 2𝑢_(5−1)+3 = 𝟐𝒖_𝟒+𝟑 = 2 × 29 + 3 = 58 + 3 = 61 Sixth term = 𝑢_6 = 2𝑢_(6−1)+3 = 𝟐𝒖_𝟓+𝟑 = 2 × 61 + 3 = 122 + 3 = 125 Seventh term = 𝑢_7 = 2𝑢_(6−1)+3 = 𝟐𝒖_𝟔+𝟑 = 2 × 125 + 3 = 250 + 3 = 253 Since sequence jumps from 125 straight to 253 Therefore, no, 133 is not a term of this sequence