find the first term of a harmonic sequence whose fifth term is 1/28 and whose 13th term is 1/178
+129840 The fifth term can be expressed as
1 / [ a + 4d] = 1/28
And the 13th term can be expressed as
1 / [ a + 12d] = 1/178
So.......this implies that
a + 4d = 28 (1) and
a + 12d = 178 (2)
Multiply (1) by -1 and add it to (2)
8d = 150 divide both sides by 8
d = 75/4
Using (1) to find a, we have
a + 4(75/4) = 28
a + 75 = 28 subtract 75 from each side
a = -47
The first term is given by 1/a = 1/ - 47 = -1/ 47
+129840 The fifth term can be expressed as
1 / [ a + 4d] = 1/28
And the 13th term can be expressed as
1 / [ a + 12d] = 1/178
So.......this implies that
a + 4d = 28 (1) and
a + 12d = 178 (2)
Multiply (1) by -1 and add it to (2)
8d = 150 divide both sides by 8
d = 75/4
Using (1) to find a, we have
a + 4(75/4) = 28
a + 75 = 28 subtract 75 from each side
a = -47
The first term is given by 1/a = 1/ - 47 = -1/ 47
Will use the reciprocal of the numbers stated:
The difference between 13th term - 5th term=8
The difference between their values: 178 - 28 =150
Therefore, the common difference is:150/8 =18.75
a(5)=F + (5 - 1).18.75
28 =F + 75
F=28 - 75 =-47. Now, we use the reciprocal of the answer:
-1/47 - The first term. Similarly,................(1)
a(13)=F + (13 -1).18.75
178 =F + 225
F=178 - 225
F=-47 or -1/47 - The first term..................(2)
