Question

What is the nth term rule of the quadratic sequence below?
6,12,20,30,42,56,72

Answer 1

The nth term rule of the given quadratic sequence is needed.

The nth term rule is $T_n=n^2+3n+2$

The quadratic sequence is

$6,12,20,30,42,56,72$

We know the sequence is quadratic so the general sequence will be of the form

$T_n=an^2+bn+c$

For $n=1,2,3$ the general terms will be

$T_1=a+b+c=6$

$T_2=4a+2b+c=12$

$T_3=9a+3b+c=20$

$T_2-T_1=3a+b\\\Rightarrow 3a+b=6$

$T_3-T_2=5a+b\\\Rightarrow 5a+b=8$

Subtracting the above equations

$T_2-T_1-(T_3-T_2)=3a+b-(5a+b)\\\Rightarrow 6-8=-2a\\\Rightarrow -2=-2a\\\Rightarrow a=\dfrac{-2}{-2}=1$

Finding $b$ and $c$

$3a+b=6\\\Rightarrow b=6-3a=6-3\times 1\\\Rightarrow b=3$

$a+b+c=6\\\Rightarrow c=6-a-b=6-1-3\\\Rightarrow c=2$

So,

$a=1$, $b=3$ and $c=2$

Substituting in $T_n$

$T_n=n^2+3n+2$

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