Grove HSC 3U EX 8.4 Q11

Hi sir when you are doing arithmetic series how do you do it when you want to know if a number fits into a series? Like Q11 from 8.4?

Answer: I would find an expression for the nth Term $T_n$.

Is $0$ a term of the series $48 + 45 + 42 + ...$?

Method 1: write them all out (only works for small numbers!)
48+45+42+39+36+33+30+27+24+21+18+15+12+9+6+3+0+...
So yes $0$ is a term.

Method 2: all the numbers are multiples of 3, and they go down by 3, so 0 must eventually be reached!! So again Yes.

Method 3: It's an AP with $a=48$, $d=-3$, 
$T_n=a+(n-1)d$
$T_n=48-3(n-1)$
$T_n=51-3n$

Solve $51-3n=0$

$51=3n$

$n=51/3=17$

$T_{17}=0$

So Yes. The $17$ th term equals $0$.

JH 2016 :-)