*5% in decimal form is 0.05.
After one week we have:
N=400*1.05 beetles
After two weeks we have:
N=(400*1.05)*1.05 beetles
After three weeks we have:
N=((400*1.05)*1.05)*1.05 beetles
We can write this as:
Week 1:
[tex]N=400* 1.05^{1} [/tex]
Week 2:
[tex]N=400* 1.05^{2} [/tex]
Week 3:
[tex]N=400* 1.05^{3} [/tex]
From this we can make conclusion that formula for number of beetles after w weeks is given by:
[tex]N=400* 1.05^{w} [/tex]
