The following sequences of numbers forms a quadratic sequences.
-3,-2,-3,-6,-11
4.1The first difference of the above sequence also form a sequence. Determine an expression for the general term of the first difference.
4.2 Calculate the first difference between the 35th and 36 terms of the quadratic sequence.
4.3 Determine an expression for the nth term of the quadratic sequence
4.4 Explain why the sequence of numbers will never contain a positive term
To determine the general term of the first difference sequence, we can subtract each term from its previous term.
-2 - (-3) = 1
-3 - (-2) = -1
-6 - (-3) = -3
-11 - (-6) = -5
So the first difference sequence is 1, -1, -3, -5.
To find the first difference between the 35th and 36th terms of the quadratic sequence, subtract the 35th term from the 36th term.
-11 - (-6) = -11 + 6 = -5
So the first difference between the 35th and 36th terms is -5.
To determine an expression for the nth term of the quadratic sequence, we can observe that the difference between each term and its previous term is increasing by 2 each time. The second difference is constant at -2.
So, the general formula for the nth term of the quadratic sequence can be given as:
Tn = an^2 + bn + c
where a is the second difference (-2), b is the first difference of the first term (-3), and c is the first term in the sequence (-3).
Therefore, the expression for the nth term of the quadratic sequence is Tn = -2n^2 - 3n - 3.
The sequence of numbers will never contain a positive term because the quadratic equation (-2n^2 - 3n - 3) has a negative leading coefficient (-2), meaning that the parabola opens downwards. Thus, all terms of the sequence will be negative.