Question

What is the nth term rule of the quadratic sequence below? − 7 , − 6 , − 3 , 2 , 9 , 18 , 29 , . .

Answer 1

Let the nth term be given by the expression an²+bn+c.

We can write A: -7=a+b+c B: -6=4a+2b+c C: -3=9a+3b+c

D: B-A: 1=3a+b
E: C-B: 3=5a+b
E-Dm 2=2a, a=1, b=1-3=-2, c=-7-1+2=-6.

So the expression for the nth term is n²-2n-6.

Put n=4: 16-8-6=2 which fits the sequence.
Put n=5: 25-10-6=9 which also fits the sequence.
So the nth term is

Answer 2

Answer:

The next term is 44.

Step-by-step explanation:

From the statement of the problem we know that the n-th term is given by a quadratic expression. So, if we denote the n-th term by $a_n$, then it can be written as $a_n = an^2+bn+c.$ Then, we have to find the coefficients $a$, $b$ and $c$.

The idea here is to obtain a system of linear equations with $a$, $b$ and $c$ as unknowns. As we need three of those equation we substitute $n=1$, $n=2$ and $n=3$ in the expression for $a_n$. Thus

$-7=a+b+c$

$-6=a2^2+b2+c = 4a+2b+c$

$-3=a3^2+b3+c = 9a+3b+c.$

The solutions of this system of linear equations is $c=-6$, $b=-2$ and $a=1$.

Hence, $a_n = n^2-2n-6.$

Notice that,

$a_4 = 4^2-2\cdot 4-6 = 16-8-6=2,$

$a_5 = 5^2-2\cdot 5-6 = 9.$

The above values correspond with ones given in the statement of the exercise. As we have 7 values given, we need to find the 8th:

$a_8 = 8^2-2\cdot 8-6 = 64-16-6=64-22 = 44,$