Consider an arithmetic sequence with common difference 20. If the sum of the reciprocals of two consecutive terms of this sequence is 1/24, what is the first term of the arithmetic sequence ?
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The terms of an arithmetic sequence with common difference 4 are natural numbers.
(a) Let x be a term of the sequence and y be the next term of the sequence.
Common difference =yx
4=yx
y=x+4
So the next term is x+4.
(b) Let x and x+4 be the two consecutive terms of sequence such that sum of their reciprocals is
15
4
.
x
1
+
x+4
1
=
15
4
x
2
+4x
2x+4
=
15
4
30x+60=4x
2
+16x
4x
2
−14x−60=0
4x
2
−24x+10x−60=0
4x(x−6)+10(x−6)=0
(4x+10)(x−6)=0
x=
2
−5
or x=6
But terms of the sequence are natural numbers.
∴x=6 and x+4=10.
The two terms of the sequence are 6 and 10.
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