2015/077) Find the formula for the number pattern : $2,5,10,17 \cdots$

The 1st order difference $3,5,7$
The second order difference $2,2,2,$
So the term is quadratic

$t_n = an^2 + bn + c$

put $n = 1$ to get

$a + b+ c = 2\cdots(1)$

put $n =2$ to get

$4a + 2b + c = 5 \cdots(2)$

put $n = 3$ to get

$9a + 3b + c = 10 \cdots(3)$

subtract (1) from (2) to get
 

$3a + b = 3 \cdots(4)$

subtract (2) from (3) to get
 

$5a + b = 5 \cdots(5)$

from (4) and (5) $2a = 2$ or $a = 1$ so $b = 0$ and hence $c = 1$
so

$t_n = n^2 + 1$