Sequence Terms 3Cafff

1. We are given two sequences: a) $$a_n = \frac{n}{n+3}$$ b) $$a_n = (-1)^n \frac{n-1}{3n+2}$$ 2. For sequence (a), the formula is $$a_n = \frac{n}{n+3}$$ which means each term is the ratio of the term index $$n$$ to $$n+3$$. 3. For sequence (b), the formula is $$a_n = (-1)^n \frac{n-1}{3n+2}$$ which includes an alternating sign factor $$(-1)^n$$ and a rational expression $$\frac{n-1}{3n+2}$$. 4. To understand the behavior, let's compute the first few terms for each sequence. 5. For (a): - $$a_1 = \frac{1}{1+3} = \frac{1}{4}$$ - $$a_2 = \frac{2}{2+3} = \frac{2}{5}$$ - $$a_3 = \frac{3}{3+3} = \frac{3}{6} = \frac{1}{2}$$ 6. For (b): - $$a_1 = (-1)^1 \frac{1-1}{3(1)+2} = -1 \times \frac{0}{5} = 0$$ - $$a_2 = (-1)^2 \frac{2-1}{3(2)+2} = 1 \times \frac{1}{8} = \frac{1}{8}$$ - $$a_3 = (-1)^3 \frac{3-1}{3(3)+2} = -1 \times \frac{2}{11} = -\frac{2}{11}$$ 7. These sequences can be graphed to visualize their behavior. 8. The alternating sign in (b) causes the terms to alternate positive and negative. 9. Both sequences involve rational expressions which approach limits as $$n$$ grows large. 10. For large $$n$$, sequence (a) approaches $$\lim_{n \to \infty} \frac{n}{n+3} = 1$$. 11. For large $$n$$, sequence (b) approaches $$\lim_{n \to \infty} (-1)^n \frac{n-1}{3n+2} = (-1)^n \frac{1}{3}$$ which oscillates between $$\frac{1}{3}$$ and $$-\frac{1}{3}$$. Final answers: - Sequence (a) terms: $$a_1=\frac{1}{4}, a_2=\frac{2}{5}, a_3=\frac{1}{2}$$ - Sequence (b) terms: $$a_1=0, a_2=\frac{1}{8}, a_3=-\frac{2}{11}$$