Question

The numbers 2/1/3,h,k,7/7/8 firm a geometric progression. find the value of h and of k

Answer 1

`\text{We have }\ 2(1)/(3),\ h,\ k,\ 7(7)/(8).\\\\\text{Convert the mixed numbers to the improper fractions}\\\\2(1)/(3)=(2\cdot3+1)/(3)=(7)/(3)\\\\7(7)/(8)=(7\cdot8+7)/(8)=(63)/(8)\\\\\text{If}\ a_1,\ a_2,\ a_3,\ a_4\ \text{is a geometric seqence, then:}\\\\(a_2)/(a_1)=(a_3)/(a_2)=(a_4)/(a_3)=r\\\\a_2=a_1r\\\\\text{and}\ (a_4)/(a_1)=r^3\\\\\text{Substitute:}`

`r^3=((63)/(8))/((7)/(3))\\\\r^3=(63)/(8)\cdot(3)/(7)\\\\r^3=(9\cdot3)/(8\cdot1)\\\\r^3=(27)/(8)\to r=\sqrt[3]{(27)/(8)}\\\\r=(3)/(2)\\\\h=2(1)/(3)\cdot(3)/(2)=(7)/(3)\cdot(3)/(2)=(7)/(2)\\\\k=(7)/(2)\cdot(3)/(2)=(21)/(4)\\\\Answer:\ \boxed{h=(7)/(2)=3(1)/(2)\ and\ k=(21)/(4)=5(1)/(4)}`