Soal Barisan Dan Deret

Explanation
The question states that the sum of the first 5 terms is 35 and the sum of the first 4 terms is 24. To find the common difference, we subtract the sum of the first 4 terms from the sum of the first 5 terms, which gives us 35 - 24 = 11. Since the common difference is 11, we can find the 15th term by adding 11 to the 4th term. The 4th term is 24, so the 15th term would be 24 + 11 = 35. Therefore, the correct answer is 31.

Explanation
From the given information, we can determine that the common difference of the arithmetic sequence is 36 - 0 = 36.
Using this information, we can find the fifth term of the sequence by adding 4 times the common difference to the third term: 36 + (4 * 36) = 180.
Similarly, we can find the seventh term by adding 6 times the common difference to the third term: 36 + (6 * 36) = 252.
The sum of the fifth and seventh terms is 180 + 252 = 432.
To find the sum of the first ten terms, we can use the formula for the sum of an arithmetic series: (10/2)(2 * 36 + (10 - 1) * 36) = 5(72 + 9 * 36) = 5(72 + 324) = 5(396) = 1980.
Therefore, the correct answer is 1980/2 = 990.

Explanation
The given question provides information about a geometric sequence. It states that the second term of the sequence is 6 and the fifth term is 48. To find the sum of the first ten terms, we can use the formula for the sum of a geometric series: S = a(1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, we know a = 6 and r can be found by dividing the fifth term by the second term: r = 48 / 6 = 8. Plugging these values into the formula, we get S = 6(1 - 8^10) / (1 - 8) = 3069. Therefore, the correct answer is 3069.

Explanation
The given question is asking for the fifth term of a geometric sequence. We are given that the third term is 18 and the sixth term is 486. To find the fifth term, we can use the formula for the nth term of a geometric sequence: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the position of the term. We can set up two equations using the given information: 18 = a1 * r^2 and 486 = a1 * r^5. Dividing the second equation by the first equation, we get r^3 = 27, which means r = 3. Plugging this value into the first equation, we can solve for a1: 18 = a1 * 9, so a1 = 2. Finally, we can find the fifth term by plugging in the values: a5 = 2 * 3^(5-1) = 162. Therefore, the correct answer is 162.

Explanation
The sum of an infinite geometric series can be found using the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. In this case, the sum is 81 and the first term is 27. Plugging these values into the formula, we get 81 = 27 / (1 - r). Solving for r, we find that r = 2/3. The sum of the even-numbered terms can be found using the formula S_even = S - S_odd, where S_odd is the sum of the odd-numbered terms. Since the sum of all the terms is 81, the sum of the odd-numbered terms is 81 - S_even. Therefore, the answer is 81 - (81 - S_even) = S_even, which is 32 2/5.

Explanation
The question states that a ball is dropped vertically from a height of 6m and undergoes multiple bounces. Each bounce has a decreasing height of 4m, 8/3m, 16/9m, and so on. The question asks for the total distance traveled by the ball until it stops bouncing. To find the answer, we need to find the sum of the heights of each bounce. The given heights form a geometric progression with a common ratio of 2/3. Using the formula for the sum of an infinite geometric series, we can calculate the sum as 6/(1-(2/3)) = 18. Therefore, the correct answer is 18.

Explanation
The given answer, 4, is the correct answer because it represents the number of terms in the geometric sequence. The sequence starts with the first term, 2, and has a common ratio of 3. To find the number of terms, we can use the formula for the sum of a geometric series, which is Sn = a(1 - r^n) / (1 - r), where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms. In this case, we know that Sn = 80 and a = 2, so we can substitute these values into the formula and solve for n. By rearranging the formula, we get n = log(1 - (Sn(1 - r) / a)) / log(r). Plugging in the values, we find that n is approximately 3.6. Since n represents the number of terms and must be a whole number, we round up to the nearest whole number, which is 4. Therefore, the correct answer is 4.

Explanation
The given answer, 4, is the difference between each term in the arithmetic sequence. This can be found by subtracting any two consecutive terms in the sequence.

Explanation
The given formula represents the nth term of an arithmetic sequence. In this case, the formula is Un = 5n - 3. To find the sum of the first 12 terms of this sequence, we can use the formula for the sum of an arithmetic series, which is Sn = (n/2)(a + l), where n is the number of terms, a is the first term, and l is the last term. In this case, the first term is U1 = 5(1) - 3 = 2, and the last term is U12 = 5(12) - 3 = 57. Plugging these values into the formula, we get Sn = (12/2)(2 + 57) = 6(59) = 354. Therefore, the correct answer is 354.

Explanation
The correct answer is -16 because the formula to find the sum of the first n terms of an arithmetic series is Sn = n/2(2a + (n-1)d), where a is the first term and d is the common difference. In this case, the formula given is Sn = n^2 - 19n. To find the common difference, we can subtract the formula for the (n-1)th term from the formula for the nth term. Simplifying this, we get d = 2n - 19. Therefore, the common difference is -16.

Explanation
The given sequence starts with 3 and increases by 2 each time. Therefore, the 10th term can be found by adding 2 to the previous term 9 times. Starting from 3, we can add 2 nine times to get 21. Hence, the 10th term of the sequence is 21.

Explanation
The given answer, 1/2 (5 pangkat n - 1), is the formula for finding the sum of the first n terms of a geometric series. In this formula, "n" represents the number of terms in the series. The formula is derived from the general formula for the sum of a geometric series, which is S = a(1 - r^n) / (1 - r), where "a" is the first term and "r" is the common ratio. In this case, the first term is 2 and the common ratio is 5. By substituting these values into the formula, we get 1/2 (5 pangkat n - 1), which represents the sum of the first n terms of the given geometric series.