How do you find the explicit formula and calculate term 20 for -1, 6, 25, 62, 123?

Oct 28, 2015

Look at sequences of differences to construct a formula and find the $20$th term is $7998$.

Explanation:

Each of the last 3 terms looks close to a cube, so I could guess the formula as ${a}_{n} = {n}^{3} - 2$, but let's pretend I didn't spot that...

Write out the initial sequence:

$\textcolor{b l u e}{- 1} , 6 , 25 , 62 , 123$

Write out the sequence of differences of that sequence:

$\textcolor{b l u e}{7} , 19 , 37 , 61$

Write out the sequence of differences of that sequence:

$\textcolor{b l u e}{12} , 18 , 24$

Write out the sequence of differences of that sequence:

$\textcolor{b l u e}{6} , 6$

Having reached a constant sequence, we can now use the first number of each sequence to write out a formula for ${a}_{n}$ as follows:

a_n = color(blue)(-1)/(0!) +color(blue)(7)/(1!) (n-1) + color(blue)(12)/(2!) (n-1)(n-2) + color(blue)(6)/(3!) (n-1)(n-2)(n-3)

$= - 1 + 7 \left(n - 1\right) + 6 \left(n - 1\right) \left(n - 2\right) + \left(n - 1\right) \left(n - 2\right) \left(n - 3\right)$

$= - 1 + 7 n - 7 + 6 {n}^{2} - 18 n + 12 + {n}^{3} - 6 {n}^{2} + 11 n - 6$

$= {n}^{3} - 2$

So ${a}_{20} = {20}^{3} - 2 = 8000 - 2 = 7998$