A geometric sequence has terms f(2)=24 and f(5)=3. What is the explicit rule for the sequence

The n'th term of a geometric sequence is given by f(n) = a*r^n-1 (assuming we start counting at n = 1), so here we have:

 

a*r^2-1 = 24  or  a*r = 24 

and

a*r^5-1 = 3 or a*r⁴ = 3

 

Divide the f(5) term by the f(2) term

a*r⁴/(a*r) = 3/24   or  r³ = 1/8  or  r³ = (1/2)³ so  r = 1/2

 

Put this back into the f(2) term to get  a*(1/2) = 24,  so a = 48.

 

Therefore  f(n) = 48*(1/2)^n-1  or f(n) = 48/2^n-1

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