4. Find the general term of the following sequence. Also, find the \( 15^{\text {th }} \) term of each ser (a) \( 8,11,14,17, \ldots \) (h) \( t_{n}=\frac{(-1)^{n}(3 n+1)}{2 n} \) (i) \( \mathrm{u}_{\mathrm{n}}=(-1)^{n}\left(\mathrm{n}^{2}+\right. \) 5. Observe the following sequences and answer the questions. (a) \( 4,7,12,19, \ldots \) (b) \( 1,7,17,31, \ldots \) (c) \( 2,1,-2,-7, \ldots \) (i) Add next three terms in each sequence.

We will treat each part in order. (Note: In part (4i) the general‐term formula appears to be incomplete; see the note at the end.) ───────────────────────────── 4. Find the general term of the following sequence and the 15th term. (a) 8, 11, 14, 17, … • Observation: This is an arithmetic sequence with first term a₁ = 8 and common difference d = 3. • General term:   aₙ = a₁ + (n – 1)d = 8 + 3(n – 1) = 3n + 5. • The 15th term:   a₁₅ = 3·15 + 5 = 45 + 5 = 50. ───────────────────────────── (h) The sequence is given by   tₙ = [ (–1)ⁿ (3n + 1) ] / (2n). Since a formula is already provided, that is the general term. To find the 15th term plug in n = 15:   t₁₅ = [ (–1)¹⁵ (3·15 + 1) ] / (2·15)      = [ (–1) (45 + 1) ] / 30     = (–46) / 30      = –23/15. ───────────────────────────── (i) The sequence is given as   uₙ = (–1)ⁿ (n² + … ) Unfortunately, the expression inside the parentheses is incomplete. Without the full expression for uₙ we cannot find a closed‐form formula or the 15th term. (Please check the original problem statement for the complete formula.) ───────────────────────────── 5. For each of the following sequences, find the next three terms. (a) 4, 7, 12, 19, … • Step 1. Find the differences:   7 – 4 = 3,   12 – 7 = 5,   19 – 12 = 7.   The differences are 3, 5, 7, … (increasing by 2 each time). • Step 2. The next differences will be 9, then 11, then 13. • Step 3. Compute the next terms:   Next term: 19 + 9 = 28.   Following: 28 + 11 = 39.   Then: 39 + 13 = 52. Thus, the next three terms are 28, 39, and 52. ───────────────────────────── (b) 1, 7, 17, 31, … • Step 1. Find the differences:   7 – 1 = 6,   17 – 7 = 10,   31 – 17 = 14.   The differences are 6, 10, 14. Notice that these increase by 4 each time. • Step 2. The next difference is: 14 + 4 = 18, then 18 + 4 = 22, and then 22 + 4 = 26. • Step 3. Compute the next terms:   Next term: 31 + 18 = 49.   Following: 49 + 22 = 71.   Then: 71 + 26 = 97. Thus, the next three terms are 49, 71, and 97. ───────────────────────────── (c) 2, 1, –2, –7, … • Step 1. Find the differences:   1 – 2 = –1,   (–2) – 1 = –3,   (–7) – (–2) = –5.   The differences are –1, –3, –5 (odd numbers with a negative sign). • Step 2. The next differences will be –7, then –9, then –11. • Step 3. Compute the next terms:   Next term: –7 + (–7) = –14.   Following: –14 + (–9) = –23.   Then: –23 + (–11) = –34. Thus, the next three terms are –14, –23, and –34. ───────────────────────────── Summary of Answers: 4(a) General term: aₙ = 3n + 5   15th term: 50 4(h) General term: tₙ = [ (–1)ⁿ (3n + 1) ]⁄(2n)  15th term: –23/15 4(i) Incomplete definition. Please verify the complete formula. 5(a) Next three terms: 28, 39, 52 5(b) Next three terms: 49, 71, 97 5(c) Next three terms: –14, –23, –34 Feel free to ask if you have any questions or need further clarification!