A) Write down and simplify the expansion of (a + b/a)^6

, for a, b ∈ R \ 0. (Hint: Use the Binomial Theorem.)

b) What is the coefficient of the b^3 term?

c) Let b = 1. What is the simplified form of the expression now?

a) a^6+6a^4b+15a^2b^2+20b^3+15(b^4/a^2)+6(b^5/a^4)+(b/a)^6

b) 20

c) a^6+6a^4+15a^2+20+15/a^2+6/a^4+1/a^6

Step-by-step explanation:

(a+b/a)^6=a^6+6a^5(b/a)+15a^4(b/a)^2+20a^3(b/a)^3+15a^2(b/a)^4+6a(b/a)^5+(b/a)^6

a^6+6a^4b+15a^2b^2+20b^3+15(b^4/a^2)+6(b^5/a^4)+(b/a)^6

b) the coefficient of b^3=20

c) if b=1, the expression is

a^6+6a^4+15a^2+20+15/a^2+6/a^4+1/a^6

constant of proportionality = 3 and the equation is c = 3p.

step-by-step explanation:

c = kp   where c = amount of caffeine in mg., p = oz of pepsi and k is the constant of proportionality.

substituting the values given:

24 = 8k

k = 3.

the same value of k will be obtained if we substitute the other given values.