Let AP(a; d) denote the set of all the terms of an infinite...

Let ${T}_{\left(1, m\right)}: {m}^{th}$ term of ${1}^{st}$ series $A\left(1; 3\right)$
${T}_{\left(2, n\right)}: {n}^{th}$ term of ${2}^{nd}$ series $A\left(2; 5\right)$
${T}_{\left(3, r\right)}: {r}^{th}$ term of ${3}^{rd}$ series $A ( 3; 7 )$
For common terms
${T}_{( 1, m )} = {T}_{( 2, n )} = {T}_{( 3, r )}$
$1 + 3\left(m - 1\right) = 2 + 5\left(n - 1\right) = 3 + 7\left(r - 1\right)$
$3m - 2 = 5n - 3 = 7r - 4$
$⇒ m = \dfrac{5n - 1}{3}andr = \dfrac{5n + 1}{7}$
For $‘m’$ to be a natural number $n = 2,5, 11, \ldots$
And for $‘r’$ to be a natural number $n = 4,11, \ldots$
When $n = 11, m = 18 and r = 8$
$⇒$ For ${1}^{st}$ common term of three series $A ( a, d )$
$a = {1}^{st}$ term of $A ( 1,3 )$
$a = 1 + ( 181 ) 3 = 52$
$⇒$ Common difference of $A\left(a, d\right) = LCM$ of common difference of \left(A\left(1; 3\right), A\left(2; 5\right), A\left(3; 7\right)\right)$<br />$d = LCM ( 3, 5, 7 )$<br />$d = 105$<br />Hence,$a + d = 52 + 105 = 157