QJEEAdv19Ma8ad0: JEE Advanced 2019 Mathematics - Class 11 JEEAdv19Ma8ad0 Solution: Let AP(a; d) denote the set of all the terms of an infinite…

Given that, AP(a ; d) denote the set of all the terms of an infinite arithmetic progression with first term 'a' and common difference d > 0.

Now, let m^th term of first progression

$$AP(1;3)=1+(m-1)3=3m-2$$ .... (i)

and n^th term of progression

$$AP(2;5)=2+(n-1)5=5m-3$$ .... (ii)

and r^th term of third progression

$$AP(3;7)=3+(r-1)7=7m-4$$ .... (iii) are equal.

Then, $$3m-2=5n-3=7r-4$$

Now, for $$AP(1;3)\cap AP(2;5)\cap AP(3;7)$$,

the common terms of first and second progressions, $$m=\frac{5n-1}{3}$$

$$\Rightarrow$$ n = 2, 5, 11, ...

and the common terms of second and the third progressions,

$$r=\frac{5n+1}{7}$$ $$\Rightarrow$$ n = 4, 11, ....

Now, the first common term of first, second and third progressions (when n = 11), so

a = 2 + (11 - 1)5 = 52

and d = LCM (3, 5, 7) = 105

So, $$AP(1;3)\cap AP(2;5)\cap AP(3;7)$$ = AP(52; 105)

So, a = 52 and d = 105

$$\Rightarrow$$ a + d = 157.00