## 1) To find the 20th term of an arithmetic sequence, we can use the formula an = a1 + (n-1)d, where an represents the nth term, a1 is the first term, and d is the common difference. In this case, a1 is 3 and d is 7. Plugging these values into the formula, we get a20 = 3 + (20-1)7 = 3 + 19*7 = 3 + 133 = 136. Therefore, the answer is b) 136.

2) To write an equation for the nth term of an arithmetic sequence, we can look for patterns in the sequence. In this case, the sequence starts with -3 and each term increases by 6. So, the common difference (d) is 6. By plugging in the first term (a1) and the common difference (d) into the formula an = a1 + (n-1)d, we get an = -3 + (n-1)6 = -3 + 6n - 6 = 6n - 9. Therefore, the answer is b) 6n + 9.

3) To find the two arithmetic means between 4 and 22, we need to find the values of a3 and a4 in the arithmetic sequence. Using the formula an = a1 + (n-1)d where a1 is the first term, d is the common difference, and n represents the position of the term, we get a3 = 4 + (3-1)d = 4 + 2d and a4 = 4 + (4-1)d = 4 + 3d. From the problem, we know that these two arithmetic means have the same value, so 4 + 2d = 4 + 3d. Subtracting 4 from both sides, we have 2d = 3d. Subtracting 2d from both sides, we get d = 0. Now, substituting the value of d into the equation 4 + 2d = 4 + 3d, we have 4 + 2(0) = 4 + 3(0), which simplifies to 4 = 4. Therefore, the means between 4 and 22 are k=4 and k=4. The answer is d) 13, 13.

4) To simplify and find Sn for an arithmetic series, we can use the formula Sn = n/2(a1 + an), where Sn represents the sum of the first n terms, a1 is the first term, and an is the nth term. In this case, a1 is 3, d is 1/2, and an is 15. Plugging these values into the formula, we get Sn = n/2(3 + 15) = n/2(18) = 9n. The answer is b) 9.

5) It seems like you are looking for the value of the expression 8 âˆ‘(k=3)^(40-3k). To solve this, we need to evaluate the expression for various values of k. Starting with k=3, we plug in this value into the expression, giving us 40 - 3(3) = 40 - 9 = 31. Now, we move on to k=4, which yields 40 - 3(4) = 40 - 12 = 28. We continue this pattern until we reach k=8, which gives us 40 - 3(8) = 40 - 24 = 16. Finally, we sum up all the resulting values: 31 + 28 + 25 + 22 + 19 + 16 = 141. Therefore, the answer is d) 141.