Question

Find the nth term of this quadratic sequence
4. 7, 12, 19, 28. . . .

Answer 1

`a_n=n^2 + 3`

Explanation:

To find the nth term of a quadratic sequence, we need to determine the quadratic function that generates the sequence.

Given quadratic sequence:

4, 7, 12, 19, 28, ...

Begin by calculating the first differences between consecutive terms:

`4 \underset{+3}{\longrightarrow}7 \underset{+5}{\longrightarrow} 12\underset{+7}{\longrightarrow} 19\underset{+9}{\longrightarrow} 28`

As the first differences are not the same, we need to calculate the second differences (the differences between the first differences):

`3\underset{+2}{\longrightarrow} 5 \underset{+2}{\longrightarrow} 7\underset{+2}{\longrightarrow}9`

As the second differences are the same, the sequence is quadratic and will contain an n² term.

The coefficient of the n² term is half of the second difference.

As the second difference is 2, the coefficient of the n² term is 1.

Now we need to compare n² with the given sequence (where n is the position of the term in the sequence).

`\begin{array}c\cline{1-6}n&1&2&3&4&5\\\cline{1-6}n^2&1&4&9&16&25\\\cline{1-6}\sf operation&+3&+3&+3&+3&+3\\\cline{1-6}\sf sequence&4&7&12&19&28\\\cline{1-6}\end{array}`

We can clearly see that the algebraic operation that takes n² to the terms of the sequence is add 3.

Therefore, the expression to find the the nth term of the given quadratic sequence is:

`\boxed{a_n=n^2 +3}`