Question

In a harmonic progression 4th term is 1/9 and 13th term is 1/27, the 7th term is

Answer 1

The 7th term of a harmonic progression is $\frac{2}{27}$.

Step-by-step explanation:

Let first term of an Arithmetic progression = a

Common difference of Arithmetic progression = d

If a, a+d, a+2d,.......are in arithmetic progression then $\frac{1}{a} ,\frac{1}{a+d} ,\frac{1}{a+2d} ,.........$ are in harmonic progression.

Given, 4th term of harmonic progression = $\frac{1}{a+3d} = \frac{1}{9}$

=> a + 4d = 9..............(i)

13th term of harmonic progression = $\frac{1}{a+12d} =\frac{1}{27}$

=> a + 12d = 27...............(ii)

Now subtract equation(i) from equation(ii),

(a+12d) - (a+4d) = 27-9 = 18

=> a + 12d - a - 4d = 18

=> 8d = 18

=> d = $\frac{18}{8} =\frac{9}{4}$

Now plug the value of d in equation(i),

a + 4 x $\frac{9}{4}$ = 9

=> a + 9 = 9

=> a = 9 - 9 = 0

7th term of arithmetic progression = a + 6d = 0 + 6 x $\frac{9}{4}$ = $\frac{27}{2}$

7th term of harmonic progression = $\frac{1}{a+6d} = \frac{2}{27}$

Hence, the 7th term of a harmonic progression is $\frac{2}{27}$.