1. **Problem Statement:** Find the number of terms and the 12th term for the sequences in problems 4, 5, and 6. 2. **Number of Terms Formula:** For a sequence from $n=a$ to $n=b$, the number of terms is given by: $$\text{Number of terms} = b - a + 1$$ 3. **12th Term Formula:** The 12th term corresponds to $n = a + 11$ (since the first term is at $n=a$). --- ### Problem 4 (a) to (f) are sigma sums, but the user asked for 4-6, so we focus on 4, 5, and 6. ### Problem 5: (a) $\sum_{n=2}^{16} x^n$ - Number of terms: $16 - 2 + 1 = 15$ - 12th term: term at $n = 2 + 11 = 13$ is $x^{13}$ (b) $\sum_{n=3}^{25} (2n^2 + 3)$ - Number of terms: $25 - 3 + 1 = 23$ - 12th term: term at $n = 3 + 11 = 14$ is $2(14)^2 + 3 = 2 \times 196 + 3 = 395$ (c) $\sum_{k=5}^{20} \frac{(-1)^k (4k - 1)}{k + 1}$ - Number of terms: $20 - 5 + 1 = 16$ - 12th term: term at $k = 5 + 11 = 16$ is $$\frac{(-1)^{16} (4 \times 16 - 1)}{16 + 1} = \frac{1 \times (64 - 1)}{17} = \frac{63}{17}$$ ### Problem 6: (a) Sequence: 4, 7, 10, 13, 16, 19, 22 - Number of terms: 7 (given) - 12th term: Since only 7 terms, 12th term does not exist. (b) Sequence: $\frac{2}{5}, \frac{6}{7}, \frac{10}{9}, \frac{14}{11}, \frac{18}{13}, \frac{22}{15}$ - Number of terms: 6 (given) - 12th term: Does not exist. (c) Sequence: $-\frac{3}{5}, \frac{5}{9}, -\frac{7}{13}, \frac{9}{17}, -\frac{11}{21}$ - Number of terms: 5 (given) - 12th term: Does not exist. (d) Sequence: 1, 5, 25, 125, 625, 3125 - Number of terms: 6 (given) - 12th term: Does not exist. (e) Sequence: $\frac{r-1}{2}, \frac{(r-1)^2}{4}, \frac{(r-1)^3}{8}, \frac{(r-1)^4}{16}, \frac{(r-1)^5}{32}, \frac{(r-1)^6}{64}$ - Number of terms: 6 (given) - 12th term: Does not exist. (f) Sequence: $a, ar, ar^2, ar^3, ar^4, ar^5, ar^6$ - Number of terms: 7 (given) - 12th term: Does not exist. --- **Summary:** - For problem 5, number of terms and 12th term are calculated. - For problem 6, number of terms are given, 12th term does not exist as sequences have fewer than 12 terms.