Harmonically progressive terms

Algebra Level 5

Let $a_1,a_2,a_3, \ldots$ be in a harmonic progression with $a_1=5$ and $a_{20}=25$ . Find the least positive integer $n$ for which $a_n<0$ .

1 solution

Aditya Tiwari
Dec 28, 2014

It's a comman basic sequence problem. The solution goes like this :-

If the given terms are in H.P then the reciprocal of these will be in A.P.

$=> \frac{1}{25} = \frac{1}{5} + 19d$ [ Where d is the common difference of the A.P]

=> On solving for d we get $d = \frac{-4}{475}$

=> Now we have to find least value of n for which $a_{n}<0$

$=> \frac{1}{5} + \frac{(n -1)(-4)}{475} < 0$

=> On solving for n, we get n>24.75

=> Therefore least integral value of n will be 25:)