Question

The third term of the sequence is 120. The fifth term is 76.8 The explicit rule for the geometric sequence is an=?(?)n-1

Answer 1

Answer:

The explicit rule of the geometric sequence

 aₙ   = 187.5 (0.8)ⁿ⁻¹

Step-by-step explanation:

Step(i):-

Given that the third term of the sequence = 120

        tₙ = a rⁿ⁻¹  

       t₃  = a r³⁻¹ = ar²

        120 = ar² ..(i)

Given that the fifth term of the given geometric sequence = 76.8

        tₙ = a rⁿ⁻¹  

       t₅  = a r⁵⁻¹ = a r⁴

   76.8  = a r⁴...(ii)

Step(ii):-

 Dividing (ii) and (i)

         $\frac{ar^{4} }{ar^{2} } = \frac{76.8}{120}$

        r²   = 0.64

       r    =√ 0.64 = 0.8

Substitute r= 0.8 in equation (i)

         120 = ar²

       120 = a(0.8)²

⇒      $a = \frac{120}{0.64} =187.5$

Step(iii):-

The explicit rule of the geometric sequence

                 aₙ  = a rⁿ⁻¹

put a= 187.5  and r = 0.8

                aₙ   = 187.5 (0.8)ⁿ⁻¹