Find the next three terms in the given sequences and calculate the sum of the arithmetic series.

Step 1: Identify the pattern. The sequence is 1, 3, 9, 27, . Each term is obtained by multiplying the previous term by 3. This is a geometric progression with a common ratio of 3. 1 × 3 = 3 3 × 3 = 9 9 × 3 = 27 Step 2: Calculate the next three terms. The term after 27 is 27 × 3 = 81. The term after 81 is 81 × 3 = 243. The term after 243 is 243 × 3 = 729. The completed sequence is 1, 3, 9, 27, 81, 243, 729. Step 1: Identify the pattern. The sequence is 1, 3, 6, 10, . Let's find the differences between consecutive terms: 3 - 1 = 2 6 - 3 = 3 10 - 6 = 4 The differences are increasing by 1 each time (2, 3, 4, ). This means the next differences will be 5, 6, 7. Step 2: Calculate the next three terms. The term after 10 is 10 + 5 = 15. The term after 15 is 15 + 6 = 21. The term after 21 is 21 + 7 = 28. The next three terms are 15, 21, 28. Step 1: Identify the series. The series is 1 + 3 + 5 + + 39. This is a sum of consecutive odd numbers. Step 2: Determine the number of terms. The n-th odd number is given by the formula 2n - 1. To find the number of terms, we set the last term equal to 2n - 1: 2n - 1 = 39 2n = 39 + 1 2n = 40 n = (40)/(2) n = 20 There are 20 terms in the series. Step 3: Calculate the sum. The sum of the first n odd numbers is given by the formula n^2. Since n = 20, the sum is: Sum = 20^2 Sum = 400 The sum is 400. Step 1: Identify the series. The series is 1 + 2 + 3 + 4 + 5 + + 98 + 99 + 100. This is the sum of the first 100 natural numbers. Step 2: Apply the formula for the sum of the first n natural numbers. The formula for the sum of the first n natural numbers is (n(n+1))/(2). In this case, n = 100. Step 3: Calculate the sum. Sum = (100(100+1))/(2) Sum = (100(101))/(2) Sum = (10100)/(2) Sum = 5050 The sum is 5050. Send me the next one 📸