Question

Determinati numarul natural n, stiind ca :
5 la puterea 1+2+3+...+2014 = 5 la puterea n+1007

Answer 1

$1+2+3+...+2014= \frac{2014 \cdot 2015}{2}=1007 \cdot 2015. \\ \\ 5^{1+2+3+...+2014}=5^{n+1007} \\ \\ 5^{1007 \cdot 2015}=5^{n+1007} \Rightarrow 1007 \cdot 2015=n+1007 \\ \\ n=1007 \cdot 2015-1007 \\ \\ n=1007(2015-1) \\ \\ n=1007 \cdot 2014 \\ \\ n=2028098$