Question

if a1,a2,a3,........ are in a harmonic progession with a1 = 5 and a20 = 25. then least positive integer n for which a nth <0 is

Answer 1

Solution :

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Given :

$a_1,a_2,a_3....$ in H.P

With  $a_1$ = 5,

&

$a_{20}$ = 25,.

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To find :

The least positive integer n for which $a_n$  < 0

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We know that,.

⇒ $a_1$ = 5

⇒ $\frac{1}{a} = 5$

⇒ a = $\frac{1}{5}$

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And also,.

⇒ $a_{20}$ = 25

⇒ $\frac{1}{a + (20-1)d} = 25$

⇒ $\frac{1}{\frac{1}{5} + (19)d} = 25$

⇒  $\frac{1}{5} + 19d = \frac{1}{25}$

⇒ $19d = \frac{1}{25} - \frac{1}{5}$

⇒ $19d = \frac{1- 5}{25}$

⇒ $19d = \frac{- 4}{25}$

⇒ $d = \frac{-4}{25(19)}$

⇒ $d = \frac{-4}{475}$

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Hence,.

We need to find the value of n for which the least $a_n \ \textless \ 0$

⇒ $\frac{1}{a + (n-1)d} < 0$

⇒ $\frac{1}{ \frac{1}{5} + (n-1)( \frac{-4}{475} ) } \ \textless \ 0$

⇒ $\frac{1}{ \frac{1}{5} + ( \frac{4-4n}{475} ) } \ \textless \ 0$

⇒ $\frac{1}{ \frac{95}{475} + \frac{4 - 4n}{475} } \ \textless \ 0$

⇒ $\frac{1}{ \frac{95 + 4 - 4n}{475} } \ \textless \ 0$

⇒ $\frac{1}{ \frac{99 - 4n}{475} } \ \textless \ 0$

⇒ $\frac{475}{99 - 4n} \ \textless \ \frac{0}{1}$

⇒ $99 - 4n \ \textless \ \frac{0}{475}$

⇒ $99 - 4n \ \textless \ 0$

⇒ $-4n \ \textless \ -99$

⇒ $n \ \textless \ \frac{-99}{-4}$

⇒ $n \ \textless \ \frac{99}{4}$

⇒ $n \ \textless \ 24.75$

As,

We know that,

⇒ n  is less than 24.75

&

⇒ n can't be in decimals,.

Hence,

⇒ n = 24,.
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                                          Hope it Helps !!

 

Answer 2

Harmonic progression

Step-by-step explanation:

Harmonic progression is reciprocal of arithmetic progression

Given $a1,a2,a3,.....$are in harmonic progression

          First term $a1=5$ and

          20th term $a20=25$

$1/5,........,1/25$(20th term)........$1/n$(nth term)

Formula for 20th term

$1/a20=1/a1+(n-1)d$

$1/a20=1/a1+1/(n-1)d$        

$1/25-1/5=19d\\1/d=-4/(19*25)\\d=-4/(25*19)$

Harmonic  progresssion for least positive integer nth (n<0)

$an<0\\a=(n-1)d<0\\1/5-[4/(19*25)](n-1)]<0\\1/5<[4(n-1)]/(19*25)\\4(n-1)>19*5\\4n-4>95\\4n>99\\n>24.75$

Harmonic progression for least positive integer nth<0 is 24.75