how ot find R(x)

Let x be the number of weeks past

P(x) be the price of commodity in the xth week
Q(x) be the demand of commodity in the xth week

P(0)=5
Q(0)=1000

P(1)=5(1+5%)=5.25
Q(1)=1000-4=996

P(x)=5(1+5%)^x
Q(x)=1000-4x

Q(x) will reach zero when x=250 and there can be no more revenue from zero demand.

R(x)=P(x)*Q(x)
= 5(1+5%)^x * (1000-4x) Ans.

Appendix:

Cumulative revenue till xth week
= P(0)*Q(0) + P(1)*Q(1) + P(2)*Q(2) + … + P(x)*Q(x)

Maximum revenue throughout the commodity life cycle (for x=0 to 249)
= P(0)*Q(0) + P(1)*Q(1) + P(2)*Q(2) + … + P(249)*Q(249)
= 5*1000 + 5(1+5%)*(1000-4) + 5(1+5%)^2*(1000-4*2) + … + 5(1+5%)^249*(1000-4*249)
= 1,665,619,475 (with the help of Excel for calculation)

Let x be the number of 5 cents increases.
Price=5+0.05x
Number sold=1000-4x

R(x)=Price x Number Sold
=(5+0.05x)(1000-4x)
=5000-20x+50x-0.2x^2
=-0.2x^2+30x+5000