Question

Let $AP(a;d)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $d>0.$ If
$AP\left(1;3\right)\cap AP\left(2;5\right)\cap AP\left(3;7\right)=AP\left(a;d\right)$
then a + d equals___

Answer 1

Answer: 157

$T_{\left(1,m\right)}=T_{\left(2,n\right)}=T_{\left(3,r\right)}$
$T_{\left(1,m\right)}$ is $m^{th}$ term of $I^{st}$ series, $T_{(2,n)}$ is $n^{th}$ term of second series and $T_{(3,r)}$ is $n^{th}$ term of third series
$\Rightarrow 1+\left(m-1\right)3=2+\left(n-1\right)5=3+\left(r-1\right)7$
For common terms of $1^{st}$ and $2^{nd}$ series
$m=\frac{5n-1}{3}\Rightarrow n=2,5,\mathrm{11....}$
For common terms of $2^{nd}$ and $3^{rd}$ series
$r=\frac{5n+1}{7}\Rightarrow n=4,11,....$
$\Rightarrow$ First common term of $1^{st},2^{nd}$ and $3^{rd}$ series (when $n=11$)
$a=2+(11-1)5=52$
$d=L.C.M.(3,5,7)=105$
$\Rightarrow a+d=157.00$