1. Given the sequence \( 6 ; 13 ; 20 ; \ldots \) a) Determine a formula for the nth term of the sequence. b) Calculate the 21 st term of this sequence. c) Determine which term of this sequence is 97 . 2. Consider this number pattern: \( 8 ; 5 ; 2 ; \ldots \) a) Calculate the \( 15^{\text {th }} \) term. b) Determine which term of this sequence is -289 . a) Given the arithmetic sequence \( 1-\mathrm{p} ; 2 \mathrm{p}-3 ; \mathrm{p}+5 ; \ldots \) b) Determine the values of the first three terms of the sequence.

We’ll solve each part step by step. ───────────────────────────── 1. For the sequence 6; 13; 20; … a) Find a formula for the nth term. • The difference between consecutive terms is:   13 – 6 = 7  and  20 – 13 = 7. • Because the common difference d = 7 and the first term a₁ = 6, the nth term of an arithmetic sequence is given by:   aₙ = a₁ + (n – 1) × d • Plug in the values:   aₙ = 6 + (n – 1) × 7 = 6 + 7n – 7 = 7n – 1 b) Calculate the 21st term. • Using aₙ = 7n – 1 with n = 21:   a₂₁ = 7 × 21 – 1 = 147 – 1 = 146 c) Determine which term of this sequence is 97. • Set aₙ = 97:   7n – 1 = 97   7n = 98   n = 14 • Thus, 97 is the 14th term. ───────────────────────────── 2. Consider the number pattern: 8; 5; 2; … a) Calculate the 15th term. • The common difference is:   5 – 8 = –3  and  2 – 5 = –3. • The nth term is:   aₙ = a₁ + (n – 1) × d = 8 + (n – 1)(–3)   Simplify: aₙ = 8 – 3(n – 1) = 8 – 3n + 3 = 11 – 3n • For n = 15:   a₁₅ = 11 – 3×15 = 11 – 45 = –34 b) Determine which term of this sequence is –289. • Set aₙ = –289:   11 – 3n = –289   Subtract 11 from both sides:   –3n = –300   Divide by –3:   n = 100 • So, –289 is the 100th term. ───────────────────────────── 3. For the arithmetic sequence: 1 – p; 2p – 3; p + 5; … a) The sequence is arithmetic if the differences between consecutive terms are equal. That is,   [(2p – 3) – (1 – p)] must equal [(p + 5) – (2p – 3)]. • Compute the first difference:   (2p – 3) – (1 – p) = 2p – 3 – 1 + p = 3p – 4 • Compute the second difference:   (p + 5) – (2p – 3) = p + 5 – 2p + 3 = –p + 8 (which is the same as 8 – p) • Set them equal:   3p – 4 = 8 – p   Add p to both sides: 4p – 4 = 8   Add 4: 4p = 12   Divide by 4: p = 3 b) Determine the values of the first three terms of the sequence. • Substitute p = 3 into each term:   First term: 1 – p = 1 – 3 = –2   Second term: 2p – 3 = 2×3 – 3 = 6 – 3 = 3   Third term: p + 5 = 3 + 5 = 8 ───────────────────────────── Summary of Answers: 1. a) aₙ = 7n – 1  b) a₂₁ = 146  c) 97 is the 14th term 2. a) The 15th term is –34  b) –289 is the 100th term 3. For p = 3, the first three terms are –2, 3, and 8.