Math alg. 2 trig.

Question

Let f(n)=(1/2)¹+(1/2)² +(1/2)³ +...+(1/2)^{n make a table for f starting with the initial term f(1)=1/2. Find a closed form for f(n).}

Answer 1

f(n) = ½ + (½)² +...+ (½)ⁿ is the sum of the first n terms of the geometric sequence ½, (½)², (½)³,... with first term a₁ = ½ and common ratio r = ½.  

 

So, f(n) = a₁(1 - rⁿ)/(1 - r)

 

            = ½(1 - (½)ⁿ)/(1 - ½)

 

            = 1 - (½)ⁿ

 

f(1) = ½

f(2) = ½ + ¼ = ¾

f(3) = ½ +  ¼ + 1/8 = 7/8  etc

Answer 2

Completely misread the problem. 

f(n)=∑_k=1^k=n(1/2^)k,

(1/2)f(n)=∑_k=2^k=n+1(1/2)^k

subtract the second equation from the first

(1/2)f(n)=(1/2)-(1/2)ⁿ⁺¹

f(n)=1-(1/2)ⁿ