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Let AP(a:d) denote the set of the all terms of an inginate arithmetic progression with first term a and common difference d gt 0 if AP (I,3)cap AP(2,5)capAP(3,7)=AP(a,d) then a+d equals...........

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Solution :`(157.00)` Given that, AP(a:d)denote the SET of all terms of an infinite arithemtic progression with first TERM 'a' and common difference `dgt0`.
Now, LET `m^th` term of first progression
and `n^th` term of progression `AP(2:5)=2+(n-1)6=5n-3` .....(ii)
and `r^th` term of THIRD progression `AP(3:7)`
`3+(r-1)7=7r-4` ........(iii)
are equal Then
`3m-2=5n-3=7r-4`
Now, for `AP(1,3)capAP(2,5)capAP(3,7)`, the common terms of first and second PROGRESSIONS , `m=(5n-1)/(3)rArr n=2,5,11`,.........and the common tems of second and the third progressions. `r=(5n+1)/(7)rArrn=4,11`,......
Now the first common term of first, second and third progressions (when `n=11`), so `a=2+(11-1)6=52 `
and `d=LCM (3,5,7)=1-5`
So, `AP(1,3)capAP(2:5)capAP(3:7)=AP(52:105)`
So, `a=52and d=105 rArr a+d=157.00`


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