The 5th term of series $\displaystyle \frac { 10 }{ 9 } ,\frac { 1 }{ 3 } \sqrt { \frac { 20 }{ 3 } } ,\frac { 2 }{ 3 } ,...$ is
- $\displaystyle \frac { 1 }{ 3 } $
-
1
- $\displaystyle \frac { 2 }{ 5 } $
- $\displaystyle \sqrt { \frac { 2 }{ 3 } } $
The terms are 10/9, (1/3)*sqrt(20/3), 2/3. Simplifying: 10/9, (sqrt(20)/3*sqrt(3)) = (2*sqrt(5))/(3*sqrt(3)), 2/3. This is a geometric progression with first term a = 10/9 and common ratio r = (sqrt(20/3)) / (10/3) = sqrt(20/3) * 3/10 = sqrt(20/3) * sqrt(9/100) = sqrt(20*9 / 3*100) = sqrt(180/300) = sqrt(3/5). The 5th term is a*r^4 = (10/9) * (3/5)^2 = (10/9) * (9/25) = 10/25 = 2/5.
By rewriting the terms to reveal their squares, the first term is the square root of 100/81, the second term is the square root of 20/27, and the third term is the square root of 4/9. The values inside the square roots (100/81, 20/27, 4/9) form a geometric progression with the first term 100/81 and a common ratio of 3/5. To find the 5th term of this sequence, we first calculate the 5th term of the internal progression as (100/81) * (3/5)^4 = (100/81) * (81/625) = 4/25. Taking the square root of 4/25 gives the 5th term of the original series as 2/5.