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

$T_{\left(1, m\right)}=T_{\left(2,n\right)}=T_{\left(3,r\right)}$
$T_{\left(1,m\right)}$ 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
$m=\frac{5n-1}{3} \Rightarrow n=2, 5, 11 ....$
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 $n = 11$)
$a = 2 + (11 - 1)5 = 52$
$d = L.C.M. (3, 5, 7) = 105$
$\Rightarrow a+d=157.00$