5, 10, 17, 26, 37, ...
I see the pattern, but I can't figure out the equation...any advice?
Hello jzellt:
You might have noticed that difference between the numbers in sequence corresponds to the odd numbers sequence starting from 5.
$\displaystyle 5,7,9,11,.....$
Now we can write it as $\displaystyle 5+0,5+5,5+5+7,5+5+7+9,...$
We can use this formula for the addition of the odd numbers:
$\displaystyle \frac{n}{2}(2a+(n-1)*d)$
Lets plug in the values:
$\displaystyle \frac{n}{2}(2(5)+(n-1)*2)$
$\displaystyle \frac{n-1}{2}(10+(n-2)*2)$ (because at n=1, there should be addition of zero i.e 5+0,5+5,....)
Hence we have the equation: $\displaystyle 5+\frac{n-1}{2}(8+2(n-1))$
Hope this helps
Hello, jzellt!
I "eyeballed" the solution . . .
$\displaystyle 5, 10, 17, 26, 37, \hdots$
Take the differences of consecutive terms.
Then take the differences of the differences ... and so on.
. . $\displaystyle \begin{array}{cccccccccc}\text{Sequence} & 5 && 10 && 17 && 26 && 37 \\
\text{1st diff.} & & 5 && 7 && 9 && 11 \\
\text{2nd diff.} & & & 2 & & 2 & & 2 \end{array}$
We see that the second differences are constant.
This indicates that the generating function is of the second degree, a quadratic.
. . That is, $\displaystyle f(n)$ contains $\displaystyle n^2.$
With that in mind, I looked the sequence again, and I saw:
. . $\displaystyle \begin{array}{c|ccc}
n & & a_n \\ \hline
1 & 5&=&2^2+1 \\
2 & 10 &=&3^2+1 \\
3 & 17 &=&4^2+1 \\
4 & 26 &=&5^2+1 \\
5 & 37 &=&6^2+1 \end{array}$
. . .$\displaystyle \begin{array}{c|c}\vdots & \vdots \\
n & \;\;(n+1)^2+1
\end{array}$
Therefore: .$\displaystyle f(n) \:=\:(n+1)^2+1 \:=\:n^2+2n+2$
. . which verifies u2_wa's result.