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Maths-

General

Easy

Question

Let $a subscript 1 end subscript comma a subscript 2 end subscript horizontal ellipsis.. a subscript 10 end subscript$ be A.P and $h subscript 1 end subscript comma h subscript 2 end subscript horizontal ellipsis h subscript 10 end subscript$ be in H.P If $a subscript 1 end subscript equals h subscript 1 end subscript equals 2$ and $a subscript 10 end subscript equals h subscript 10 end subscript equals 3$ the $a subscript 3 end subscript h subscript 8 end subscript equals$

2/3
4
9
6

hint Hint:

The harmonic progression is formed by taking the reciprocal of the terms of the arithmetic progression. If the given terms of the arithmetic progression are a, a + d, a + 2d, a + 3d, ...., then the terms of the harmonic progression (or harmonic sequence) are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),...... Here a is the first term and d is a common difference. Both a and d have non-zero values.

The correct answer is: 6

Given : $a subscript 1 end subscript comma a subscript 2 end subscript horizontal ellipsis.. a subscript 10 end subscript$ are in AP and $h subscript 1 end subscript comma h subscript 2 end subscript horizontal ellipsis h subscript 10 end subscript$ are in HP
Also : $a subscript 1 end subscript equals h subscript 1 end subscript equals 2$ and $a subscript 10 end subscript equals h subscript 10 end subscript equals 3$
To Find : $a subscript 3 h subscript 8$
$a subscript 10 space equals space end subscript a subscript 1 space plus space 9 d space................ space left parenthesis space a subscript n space equals space space a space plus left parenthesis n minus 1 right parenthesis d right parenthesis h e r e space a subscript 1 space equals space space 2 space a n d space a subscript 10 space equals space 3. space S u b s t i u t i n g space t h e s e space v a l u e s rightwards double arrow 2 space plus space 9 d space space equals space 3 rightwards double arrow 9 d space equals space 1 rightwards double arrow d space equals space 1 over 9 space N o w comma space a subscript 3 space equals space a space plus space 2 d space.................. space left parenthesis space a subscript n space equals space space a space plus left parenthesis n minus 1 right parenthesis d right parenthesis S u b s t i t u t i n g space t h e space v a l u e s a subscript 3 space equals space 2 space plus space 2 cross times 1 over 9 space equals space 20 over 9 space.............. left parenthesis 1 right parenthesis$
We know that in Harmonic Progression the terms of the harmonic progression (or harmonic sequence) are the reciprocal of the terms of the arithmetic progression.
If, $h subscript 1 end subscript comma h subscript 2 end subscript horizontal ellipsis h subscript 10 end subscript$ are in HP
$rightwards double arrow 1 over h subscript 1 space comma space 1 over h subscript 2 comma space......... space comma space 1 over h subscript 10 space space a r e space i n space A P h subscript 1 space equals space space 2 space a n d h subscript 10 space equals space 3 rightwards double arrow 1 over h subscript 10 space equals space 1 over h subscript 1 space plus space 9 d subscript 1 S u b s t i t u t i n g space t h e space v a l u e s comma space w e space g e t rightwards double arrow 1 third equals space 1 half space plus space 9 d subscript 1 rightwards double arrow d subscript 1 space equals space minus 1 over 54 N o w comma space 1 over h subscript 8 space equals space 1 over h subscript 1 space plus space 7 d subscript 1 S u b s t i t u t i n g space t h e space v a l u e s comma space w e space g e t rightwards double arrow 1 over h subscript 8 space equals space 1 half space plus space 7 open parentheses negative 1 over 54 close parentheses rightwards double arrow 1 over h subscript 8 space equals space 1 half space minus space 7 over 54 rightwards double arrow 1 over h subscript 8 space equals space 20 over 54 space equals space 10 over 27 rightwards double arrow h subscript 8 space equals space 27 over 10 space......... space left parenthesis 2 right parenthesis F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis a subscript 3 h subscript 8 space equals space 20 over 9 cross times space 27 over 10 space equals space 6$

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