Question

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 ?

Answer 1

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.