Given the quadratic sequence: p;5;q;19; ....
1.1 Determine the values p and q if the second constant difference is 2.
1.2 Determine the nth term of the quadratic sequence.
1.3. Determine the first term of the sequence that will have a value greater than 10301.
1.1 to find the values of p and q, we need to determine the first and second differences of the sequence.
The first differences are: 5 - p, q - 5, 19 - q, ...
The second differences are: (q - 5) - (5 - p), (19 - q) - (q - 5), ...
Since the second constant difference is 2, we have:
(q - 5) - (5 - p) = 2
19 - q - (q - 5) = 2
Simplifying these equations, we get:
q - 5 - 5 + p = 2
19 - q - q + 5 = 2
q + p - 10 = 2
-2q + 24 = 2
q + p = 12 (equation 1)
-2q = -22
q = 11
Substituting q = 11 into equation 1, we have:
11 + p = 12
p = 1
Therefore, the values of p and q are p = 1 and q = 11.
1.2 The nth term of the quadratic sequence can be found using the formula:
An = a + (n - 1)d + (n - 1)(n - 2)/2 * c
In this case, the first term a is p = 1, the common difference d is 5 - p = 5 - 1 = 4, and the second constant difference c is 2.
So, the nth term An can be calculated as:
An = 1 + (n - 1)4 + (n - 1)(n - 2)/2 * 2
An = 1 + 4n - 4 + n^2 - 3n + 2
An = n^2 + n - 1
Therefore, the nth term of the quadratic sequence is n^2 + n - 1.
1.3 To find the first term of the sequence that will have a value greater than 10301, we need to solve the following inequality:
n^2 + n - 1 > 10301
Rearranging the inequality, we get:
n^2 + n - 1 - 10301 > 0
n^2 + n - 10302 > 0
Now, we can solve this quadratic inequality by factoring:
(n - 101)(n + 102) > 0
The solutions to this inequality are n > 101 or n < -102.
However, since we are looking for the first positive term greater than 10301, the first term will be n = 102 + 1 = 103.
Therefore, the first term of the sequence that will have a value greater than 10301 is the 103rd term.