Answer:
The sequence 3, 6, 11, 18, 27 does not match the options provided in the question. The correct nth term for this quadratic sequence should be 2n^2 + 2n - 1, which is not listed among the options (a), (b), (c), or (d).
Explanation:
To find the nth term of a quadratic sequence, you need to determine a pattern that fits all given terms of the sequence. The provided sequence is 3, 6, 11, 18, 27. If we look for a pattern, we can see that the second difference is constant, which confirms it's a quadratic sequence.
First, let's find the differences between the terms:
6 - 3 = 3
11 - 6 = 5
18 - 11 = 7
27 - 18 = 9
Now, let's find the second difference:
5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
The second difference is constant and equal to 2, which is typical for a sequence that can be represented by n^2. However, we need to check which option correctly fits all the terms given:
For first term (n = 1): n^2 = 1^2 = 1, which doesn't fit.
For second term (n = 2): n^2 = 2^2 = 4, which doesn't fit.
Now, check the term n^2 + 2n derived from the series expansions.
For first term (n = 1): 1^2 + 2(1) = 3, which fits.
For second term (n = 2): 2^2 + 2(2) = 8, which doesn't fit.
For third term (n = 3): 3^2 + 2(3) = 15, which doesn't fit either.
The correct nth term should be 2n^2 + 2n - 1. None of the provided options exactly matches this term. Therefore, the correct nth term for this quadratic sequence is not listed among the options provided of (a), (b), (c), and (d).