Number of terms=[common difference - first term + last term] / common difference
N =[1-14+180 / 1 = 167 terms
You sum them up as Arithmetic Series: S ={first + last} x number of terms / 2={14+180} x 167/2
=16,199
I think the 5 year old Gauss did the same problem but added up 1 to 100 in his head with this method.
+26387 sum of the numbers starting from 14 to 180?
\(\small{ \begin{array}{|rcll|} \hline && 14+15+16 +\cdots +178 +179+180 \\ &=& (14-13+13) + (15-13+13) + (16-13+13) +\cdots + (178-13+13) + (179-13+13) + (180-13+13) \\ &=& (1+13) + (2+13) + (3+13) +\cdots + (165+13) + (166+13) + (167+13) \\ &=& 167\cdot 13 + (1+2+3+4+\cdots + 165+166+167)\\ &=& 167\cdot 13 + \frac{1+167}{2}\cdot 167 \\ &=& 2171 + 14028 \\ &=& 16199 \\ \hline \end{array} } \)
