Question

Find the 7th term of harmonic sequence whose 2nd term is 2 and 5th term is - 1/2.

Answer 1

so, the corresponding AP is
? 1/2 ? ? -2
3d = -5/2, so d = -5/6 and the AP is
4/3, 1/2, -1/3, -7/6, -2, -17/6, -11/3
so the 7th term of the HS is -3/11

Answer 2

In harmonic sequence:

a1 = 1 / a

a2 = 1 / ( a + d )

a3 = 1 / ( a + 2 d )

a4 = 1 / ( a + 3 d )

a5 = 1 / ( a + 4 d )

a6 = 1 / ( a + 5 d )

a7 = 1 / ( a + 6 d )

in this case:

a2 = 1 / ( a + d ) = 2

a5 = 1 / ( a + 4 d ) = - 1 / 2

Second term:

1 / ( a + d ) = 2

Multiply both sides by ( a + d )

1 = 2 ( a + d )

1 = 2 a + 2 d

2 a + 2 d = 1

Fifth term:

1 / ( a + 4 d ) = - 1 / 2

Multiply both sides by ( a + 4 d )

1 = - ( a + 4 d ) / 2

Multiply both sides ba 2

2 = - ( a + 4 d )

Multiply both sides by - 1

- 2 = a + 4 d

a + 4 d = - 2

Now you must solve system:

2 a + 2 d = 1

a + 4 d = - 2
__________

The solution is:

a = 4 / 3 , d = - 5 / 6

a7 = 1 / ( a + 6 d ) = 1 / [ 4 / 3 + 6 ∙ ( - 5 / 6 ) ] = 1 / ( 4 / 3 - 5 ) =

1 / ( 4 / 3 - 15 / 3 ) = 1 / ( - 11 / 3 ) = - 3 / 11