I'm currently going through some questions on series and stumbled on this.
The sum of three consecutive terms of an Arithmetic progression is $18$ and their product is $120$. Find the terms.
I attempted it like this but I think I'm wrong cause the products is giving me $210$.
$x + (x+1) +(x+2) = 18$;
$3x +3 = 18;$
$3x = 15;$
$x=5,$ The three consecutive numbers are
$5, 6, 7$;
Appreciate the assistance.
Best Answer
In general, the consecutive terms can differ by any number, say $d$, instead of just $1$. So, the three numbers are better represented as $x-d,x,$ and $x+d$.
Then, $$3x=18 \Rightarrow x=6$$ and $$x(x^2-d^2)=120$$ $$6(36-d^2)=120$$ $$36-d^2=20$$ $$d^2=16$$ $$d=\pm4$$
The numbers thus are $\{2,6,10\}$ or $\{10,6,2\}$.