Certainly! Let’s find the values of the indicated terms for each given sequence:
1. For the sequence (a_n = {3, 7, 11, 15, \dots}):
- (a_9): To find the 9th term, we can use the formula (a_n = 4n - 1). Plugging in (n = 9): [a_9 = 4 \cdot 9 - 1 = 35]
- (a_{11}): Using the same formula, we get: [a_{11} = 4 \cdot 11 - 1 = 43]
- (a_{14}): [a_{14} = 4 \cdot 14 - 1 = 55]
2. For the sequence (a_n = {5, 7, 11, 19, \dots}):
- (a_9): Let’s find the common difference between consecutive terms: (7 - 5 = 2). The general formula for this sequence is (a_n = 5 + 2(n - 1)). Plugging in (n = 9): [a_9 = 5 + 2(9 - 1) = 21]
- (a_{10}): [a_{10} = 5 + 2(10 - 1) = 23]
- (a_{14}): [a_{14} = 5 + 2(14 - 1) = 31]
3. For the sequence (0, 2, 6, 14, 30, 62, \dots):
- To find (a_6), we can observe that each term is obtained by doubling the previous term and adding 2: [a_6 = 2 \cdot a_5 + 2 = 2 \cdot 30 + 2 = 62]
- Similarly, for (a_9): [a_9 = 2 \cdot a_8 + 2 = 2 \cdot 62 + 2 = 126]
- For the sequence (3, 8, 15, 24, \dots):
4. The common difference between consecutive terms is (8 - 3 = 5). The general formula is (a_n = 3 + 5(n - 1)).
- (a_9): [a_9 = 3 + 5(9 - 1) = 43]
- (a_{10}): [a_{10} = 3 + 5(10 - 1) = 48]
Hope it helps :)
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If I have any mistakes, I'm so sorry for that.
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