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