`6, 15, 30, 51, 78, 111.....` Decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the...

Thursday, April 13, 2017

$\displaystyle{6},{15},{30},{51},{78},{111}\ldots..$ Decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the...

Firstly we need to determine whether the series is linear or quadratic. A linear sequence is a sequence of numbers in which there is a first difference between any consecutive terms is constant. However, a quadratic sequence is a sequence of numbers in which there is a second difference between any consecutive terms is constant.

Let's begin finding the solution by finding the first difference:

$\displaystyle{x}_{{1}}={T}_{{2}}-{T}_{{1}}={15}-{6}={9}$

$\displaystyle{x}_{{2}}={T}_{{3}}-{T}_{{2}}={30}-{15}={15}$

$\displaystyle{x}_{{3}}={T}_{{4}}-{T}_{{3}}={51}-{30}={21}$

$\displaystyle{x}_{{4}}={T}_{{5}}-{T}_{{4}}={78}-{51}={27}$

From above we can see we do not have a constant first difference, now let's find out if we have a second difference:

$\displaystyle{x}_{{2}}-{x}_{{1}}={15}-{9}={6}$

$\displaystyle{x}_{{3}}-{x}_{{2}}={21}-{15}={6}$

$\displaystyle{x}_{{4}}-{x}_{{3}}={27}-{21}={6}$

From above we have a second difference, therefore we have a constant second difference. The sequence is quadratic.

Now we know our sequence is quadratic, let's find the the model using the following equation:

$\displaystyle{T}_{{n}}={a}{{n}}^{{2}}+{b}{n}+{c}$

We need to find the variables a, b, c using the following equations:

$\displaystyle{2}{a}=$ second difference therefore

$\displaystyle{2}{a}={6}$

$\displaystyle{a}={3}$

$\displaystyle{3}{a}+{b}=$ first difference between term 2 and term 1

$\displaystyle{3}{a}+{b}={9}$

$\displaystyle{3}{\left({3}\right)}+{b}={9}$ (substitute 3 for a)

$\displaystyle{b}={9}-{9}={0}$

Lastly:

$\displaystyle{a}+{b}+{c}$ = first term

$\displaystyle{3}+{0}+{c}={6}$ (substitute 3 for a and 0 for b)

$\displaystyle{c}={6}-{3}={3}$

Now we have determined all three variables, lets determine the model:

$\displaystyle{T}_{{n}}={3}{{n}}^{{2}}+{0}{n}+{3}={3}{{n}}^{{2}}+{3}$

Now we have a model, let's double check if it is correct using term 2 and term 6:

$\displaystyle{T}_{{2}}={3}{{\left({2}\right)}}^{{2}}+{3}={15}$

$\displaystyle{T}_{{6}}={3}{{\left({6}\right)}}^{{2}}+{3}={111}$

SUMMARY:

Sequence: Quadratic

Model: $\displaystyle{T}_{{n}}={3}{{n}}^{{2}}+{3}$

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