Find 3 geometric means between 8/648
Answer (1)
SOLUTION:Step 1: List the given values.Since we must insert 3 geometric means between 8 and 648, the number of terms (3 geometric means + 2 given terms) is 5.[tex]\begin{aligned} & a_1 = 8 \\ & a_5 = 648 \\ & n = 5 \end{aligned}[/tex]Step 2: Determine the common ratio.[tex]\begin{aligned} a_n & = a_1r^{n - 1} \\ a_5 & = 8r^{5 - 1} \\ 648 & = 8r^4 \\ \frac{648}{8} & = \frac{8}{8}r^4 \\ 81 & = r^4 \\ r^4 & = 81 \\ \sqrt[4]{r^4} & = \sqrt[4]{81} \\ r & = 3 \end{aligned}[/tex]Step 3: Solve for the 3 geometric means.• For a₂[tex]\begin{aligned} a_2 & = a_1r^{2 - 1} \\ a_2 & = 8(3)^1 \\ a_2 & = 8(3) \\ & = \boxed{24} \end{aligned}[/tex]• For a₃[tex]\begin{aligned} a_3 & = a_1r^{3 - 1} \\ a_3 & = 8(3)^2 \\ a_3 & = 8(9) \\ & = \boxed{72} \end{aligned}[/tex]• For a₄[tex]\begin{aligned} a_4 & = a_1r^{4 - 1} \\ a_4 & = 8(3)^3 \\ a_4 & = 8(27) \\ & = \boxed{216} \end{aligned}[/tex]Hence, the 3 geometric means between 8 and 648 are 24, 72, and 216, respectively.
