Arithmetic & Geometric Series and Sequences - Precal Adv

The series 11, 15, 19, ..., 83 is an arithmetic series with first term 11, last term 83, and common difference 4. The number of terms is 19. The sum is (19/2) * (11 + 83) = 893, making 893 the correct answer.

The series 50 + 48 + 46 + ... + 10 is an arithmetic series with first term 50, last term 10, and common difference -2. The number of terms is 21. The sum is (21/2) * (50 + 10) = 630, which is the correct answer.

The series is arithmetic with first term a = 11 and common difference d = 7. The sum of the first n terms is S_n = n/2 * (2a + (n-1)d). For n=7, S_7 = 7/2 * (2*11 + 6*7) = 224. Thus, the correct answer is 224.

The series is arithmetic with first term a=14 and common difference d=9. The sum of the first n terms is S_n = n/2 * (2a + (n-1)d). For n=11, S_11 = 11/2 * (2*14 + 10*9) = 649.

To find S_n for the arithmetic series, use S_n = n/2 * (2a_1 + (n-1)d). Here, S_21 = 21/2 * (2*22 + 20*(-8)) = 21/2 * (44 - 160) = 21/2 * (-116) = -1218. Thus, the correct answer is -1218.

To find the common difference, subtract the first term from the second: -29 - (-34) = 5. This is consistent for the next pairs: -24 - (-29) = 5 and -19 - (-24) = 5. Thus, the common difference d = 5.

The sequence is arithmetic because the difference between consecutive terms is constant: 9 - 17 = -8, 1 - 9 = -8, and -7 - 1 = -8. Since the common difference is the same, the answer is Yes.

To find the common difference, subtract the second term from the first: 97 - 86 = 11. Then, subtract the third term from the second: 86 - 75 = 11. Finally, subtract the fourth term from the third: 75 - 64 = 11. Since the sequence decreases, the common difference is -11.

The sequence is an arithmetic series with a first term of 28 and a common difference of 6. The sum of an arithmetic series can be calculated using the formula S_n = n/2 * (first term + last term). Here, S_11 = 11/2 * (28 + 88) = 638.