The pattern in this sequence is to multiply the previous term by 2 and then add 3. Let's check: \(4 \times 2 + 3 = 8 + 3 = 11\). \(11 \times 2 + 3 = 22 + 3 = 25\). \(25 \times 2 + 3 = 50 + 3 = 53\). To find the next term, we apply the rule to 53: \(53 \times 2 + 3 = 106 + 3 = 109\).

The pattern for this sequence is \(n^2 + 1\), where n is the term's position in the sequence (starting with n=1). The first term is \(1^2 + 1 = 2\). The second term is \(2^2 + 1 = 5\). The third term is \(3^2 + 1 = 10\). The fourth term is \(4^2 + 1 = 17\). The fifth term is \(5^2 + 1 = 26\). The next term (the sixth term) is \(6^2 + 1 = 36 + 1 = 37\).

The pattern for this sequence is \(n^3 - 1\), where n is the term's position in the sequence (starting with n=1). The first term is \(1^3 - 1 = 0\). The second is \(2^3 - 1 = 8 - 1 = 7\). The third is \(3^3 - 1 = 27 - 1 = 26\). The fourth is \(4^3 - 1 = 64 - 1 = 63\). The fifth is \(5^3 - 1 = 125 - 1 = 124\). The next term will be the sixth term: \(6^3 - 1 = 216 - 1 = 215\).

In an arithmetic sequence, the difference between consecutive terms is constant. The difference is \(-15 - (-21) = -15 + 21 = 6\). Let's check with the last two terms: \(3 - (-3) = 3 + 3 = 6\). The common difference is 6. To find the missing term, add 6 to the term before it: \(-15 + 6 = -9\). We can check this: \(-9 + 6 = -3\), which is correct.

The differences between consecutive terms are increasing. The differences are: \(100 - 99 = 1\), \(99 - 96 = 3\), \(96 - 91 = 5\), \(91 - 84 = 7\). The amounts being subtracted are consecutive odd numbers. The next odd number to subtract is 9. Therefore, the next term is \(84 - 9 = 75\).

First, find the common ratio \(r\) by dividing a term by its preceding term: \(r = 72 \div 108 = \frac{2}{3}\). To find the first term, we must reverse the process. Instead of multiplying by the ratio to get the next term, we divide the second term by the ratio to get the first term. \(108 \div \frac{2}{3} = 108 \times \frac{3}{2} = 162\).

The rule is to subtract the second of two consecutive terms from the first to get the next term. Let's check the given sequence: \(20 - 12 = 8\); \(12 - 8 = 4\); \(8 - 4 = 4\). To find the next term, we apply the rule to the last two terms: \(4 - 4 = 0\).

This describes an arithmetic sequence where the first term is \(a_1 = 8\) and the common difference is \(d = 6\). We need to find the 7th term, \(a_7\). Using the formula \(a_n = a_1 + (n-1)d\), we get \(a_7 = 8 + (7-1) \times 6 = 8 + 6 \times 6 = 8 + 36 = 44\). So, there are 44 dots in the 7th ring.

The sequence follows a repeating pattern of two operations: add 10, then subtract 5. \(10 + 10 = 20\); \(20 - 5 = 15\); \(15 + 10 = 25\); \(25 - 5 = 20\); \(20 + 10 = 30\). The next step is to subtract 5: \(30 - 5 = 25\).

The pattern alternates between two operations: subtracting 2 and multiplying by 3. \(5 - 2 = 3\), \(3 \times 3 = 9\), \(9 - 2 = 7\), \(7 \times 3 = 21\), \(21 - 2 = 19\). The next operation is to multiply by 3: \(19 \times 3 = 57\).