Let f and g be functions satisfying f(x+y)=f(x) f(y), f(1)=7 and g(x+y)=g(x y), g(1)=1 , for all x, y N . If _ x=1 ^ n ( f(x) g (x) )=19607 , then n is equal to:

Step-by-step solution

f(x+y)=f(x)f(y) with f(1)=7 gives f(x)=7^x . For g : put y=1 in g(x+y)=g(xy) to get g(x+1)=g(x) . So g(x)=g(1)=1 for all x N . _ x=1 ^ n f(x) g(x) = _ x=1 ^ n 7^x= 7(7^n-1) 6 =19607 7^ n+1 -7=117642 7^ n+1 =117649=7^6 n+1=6 n=5