Let A P   a ; d denote the set of all the terms of an infinite arithmetic progression with first term a…

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 AP1;3∩AP2;5∩AP3;7=APa;d then a+d equals ____

Solution

Let T1, m: mth term of 1st series A1;3
T2, n: nth term of 2nd series A2;5
T3, r: rth term of 3rd series A(3;7)
For common terms
T(1, m)=T(2, n)=T(3, r)
1+3m-1=2+5n-1=3+7r-1
3m-2=5n-3=7r-4
⇒m=5n-13   and   r=5n+17
For ‘m’ to be a natural number n=2,5,11,…
And for ‘r’ to be a natural number n=4,11,…
When n=11,m=18 and r=8
⇒ For 1st common term of three series A(a,d)
a=1st term of A(1,3)
a=1+(181)3=52
⇒ Common difference of Aa,d=LCM of common difference of A1;3,A2;5,A3;7
d=LCM (3, 5, 7)
d=105
Hence, a+d=52+105=157

Asked in: JEE Advanced 2019 (Paper 1)

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