x

### The Problem:

Find the next three terms in this sequence

14, 24, 36, 50, 66, ..., ..., ...,

-/|\-

### Solution:

### Solution

As with any sequence we are given first let's look to see if it is not one we know. It's not one I know, so let's find the first difference between adjacent terms.

$n$ | 1 | 2 | 3 | 4 | 5 | |||||

${n}^{\mathrm{th}}$
term $\left({U}_{n}\right)$ |
14 | 24 | 36 | 50 | 66 | |||||

First difference
$({U}_{n+1}-{U}_{n})$ |
10 | 12 | 14 | 16 |

This first difference is an arithmetic sequence with a common difference of 2, which enables us to work out the next three first differences:

$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||||||

${n}^{\mathrm{th}}$ term $\left({U}_{n}\right)$ |
14 | 24 | 36 | 50 | 66 | ? | ? | ? | ||||||||

First difference $({U}_{n+1}-{U}_{n})$ |
10 | 12 | 14 | 16 | 18 | 20 | 22 |

Adding these first differences in term to the previous term in the sequence we get:

$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||||||

${n}^{\mathrm{th}}$ term $\left({U}_{n}\right)$ | 14 | 24 | 36 | 50 | 66 | 84 | 104 | 126 | ||||||||

First difference $({U}_{n+1}-{U}_{n})$ |
10 | 12 | 14 | 16 | 18 | 20 | 22 |

So the sequence is 14, 24, 36, 50, 66, 84, 104, 126,…

We can go further and find any term in the sequence. The first difference is an arithmetic sequence with a common difference of 2, implying that the original sequence is a quadratic sequence of the form ${U}_{n}=a{n}^{2}+bn+c$, where a is the common difference divided by 2, ∴ $a=1$.

$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||||||

${n}^{\mathrm{th}}$ term $\left({U}_{n}\right)$ |
14 | 24 | 36 | 50 | 66 | 84 | 104 | 126 | ||||||||

First difference $({U}_{n+1}-{U}_{n})$ |
10 | 12 | 14 | 16 | 18 | 20 | 22 | |||||||||

$a{n}^{2}$ | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | ||||||||

${U}_{n}-a{n}^{2}$ | 13 | 20 | 27 | 34 | 41 | 48 | 55 | 62 |

But ${U}_{n}-a{n}^{2}=bn+c$, so the last line is an arithmetic sequence with common difference $b=7$ and zeroth term $c=6$. Therefore, the ${n}^{\mathrm{th}}$ term of the sequence is given by ${U}_{n}={n}^{2}+7n+6$.

-oOo-