Geom Prog Terms 0Ad1F2

1. **Problem 1(a): Find the first term of the geometric progression given by** $$T_n = -\frac{3}{7} r^{n-1}$$ 2. **Formula and explanation:** - The first term $$T_1$$ is found by substituting $$n=1$$ into the formula: $$T_1 = -\frac{3}{7} r^{1-1} = -\frac{3}{7} r^0 = -\frac{3}{7} \times 1 = -\frac{3}{7}$$ 3. **Answer:** - The first term $$T_1 = -\frac{3}{7}$$ 4. **Problem 1(b)(i): Find the value of $$p$$ given $$r \neq p$$** - Since $$r \neq p$$ is given but no explicit value for $$p$$ is provided, we interpret $$p$$ as a parameter distinct from $$r$$. - Without additional information, $$p$$ remains undefined. 5. **Problem 4(a): Given $$S_n = \frac{1}{2}(11n^2 + 9n)$$, find:** (a) The first term $$T_1$$ (b) The common difference $$d$$ of the arithmetic progression (c) The sum of terms from the 5th to the 9th term 6. **Step 1: Find the first term $$T_1$$** - Recall that $$S_n$$ is the sum of the first $$n$$ terms. - The first term $$T_1 = S_1$$ - Calculate: $$S_1 = \frac{1}{2}(11(1)^2 + 9(1)) = \frac{1}{2}(11 + 9) = \frac{1}{2} \times 20 = 10$$ 7. **Step 2: Find the second term $$T_2$$** - $$T_2 = S_2 - S_1$$ - Calculate $$S_2$$: $$S_2 = \frac{1}{2}(11(2)^2 + 9(2)) = \frac{1}{2}(11 \times 4 + 18) = \frac{1}{2}(44 + 18) = \frac{1}{2} \times 62 = 31$$ - Then: $$T_2 = 31 - 10 = 21$$ 8. **Step 3: Find the common difference $$d$$** - The common difference $$d = T_2 - T_1 = 21 - 10 = 11$$ 9. **Step 4: Find the sum of terms from the 5th to the 9th term** - Sum from 5th to 9th term is: $$S_9 - S_4$$ - Calculate $$S_9$$: $$S_9 = \frac{1}{2}(11(9)^2 + 9(9)) = \frac{1}{2}(11 \times 81 + 81) = \frac{1}{2}(891 + 81) = \frac{1}{2} \times 972 = 486$$ - Calculate $$S_4$$: $$S_4 = \frac{1}{2}(11(4)^2 + 9(4)) = \frac{1}{2}(11 \times 16 + 36) = \frac{1}{2}(176 + 36) = \frac{1}{2} \times 212 = 106$$ - Therefore: $$\text{Sum} = S_9 - S_4 = 486 - 106 = 380$$ **Final answers:** - (a) First term $$T_1 = 10$$ - (b) Common difference $$d = 11$$ - (c) Sum of terms from 5th to 9th term = $$380$$