Question

The terms of an arithmetic sequence with common difference 4 are natural numbers.
a) If $x$ is a term in this sequence, what is the next term?
b) If the sum of reciprocals of two consecutive terms of this sequence is $\frac{4}{15},$ find those terms.

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 $\frac{4}{15}.$
$\frac{1}{x}+\frac{1}{x+4}=\frac{4}{15}$
$\frac{2x+4}{x^2+4x}=\frac{4}{15}$
$30x+60=4x^2+16x$
$4x^2-14x-60=0$
$4x^2-24x+10x-60=0$
$4x(x-6)+10(x-6)=0$
$(4x+10)(x-6)=0$
$x=\frac{-5}{2}$ or $x=6$
But terms of the sequence are natural numbers.
$\therefore x=6$ and $x+4=10.$
The two terms of the sequence are 6 and 10.