To find the first five terms of each sequence, we need to substitute \( n = 1, 2, 3, 4, 5 \) into the given nth term formulas. This will give us the first five terms for each sequence.
To find the first five terms of the sequence defined by \(2n^2 + 5\), we substitute \(n = 1, 2, 3, 4, 5\):
- For \(n = 1\): \(2(1^2) + 5 = 7\)
- For \(n = 2\): \(2(2^2) + 5 = 13\)
- For \(n = 3\): \(2(3^2) + 5 = 23\)
- For \(n = 4\): \(2(4^2) + 5 = 37\)
- For \(n = 5\): \(2(5^2) + 5 = 55\)
Thus, the first five terms are \([7, 13, 23, 37, 55]\).
Next, we find the first five terms of the sequence defined by \(6n + 3\):
- For \(n = 1\): \(6(1) + 3 = 9\)
- For \(n = 2\): \(6(2) + 3 = 15\)
- For \(n = 3\): \(6(3) + 3 = 21\)
- For \(n = 4\): \(6(4) + 3 = 27\)
- For \(n = 5\): \(6(5) + 3 = 33\)
Thus, the first five terms are \([9, 15, 21, 27, 33]\).
Now, we calculate the first five terms of the sequence defined by \(10^n - 2\):
- For \(n = 1\): \(10^1 - 2 = 8\)
- For \(n = 2\): \(10^2 - 2 = 98\)
- For \(n = 3\): \(10^3 - 2 = 998\)
- For \(n = 4\): \(10^4 - 2 = 9998\)
- For \(n = 5\): \(10^5 - 2 = 99998\)
Thus, the first five terms are \([8, 98, 998, 9998, 99998]\).
Finally, we find the first five terms of the sequence defined by \(3n - 2\):
- For \(n = 1\): \(3(1) - 2 = 1\)
- For \(n = 2\): \(3(2) - 2 = 4\)
- For \(n = 3\): \(3(3) - 2 = 7\)
- For \(n = 4\): \(3(4) - 2 = 10\)
- For \(n = 5\): \(3(5) - 2 = 13\)
Thus, the first five terms are \([1, 4, 7, 10, 13]\).
The first five terms for each sequence are:
- For \(2n^2 + 5\): \(\boxed{[7, 13, 23, 37, 55]}\)
- For \(6n + 3\): \(\boxed{[9, 15, 21, 27, 33]}\)
- For \(10^n - 2\): \(\boxed{[8, 98, 998, 9998, 99998]}\)
- For \(3n - 2\): \(\boxed{[1, 4, 7, 10, 13]}\)
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