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 ( 1 ; 3 ) AP ( 2 ; 5 ) AP ( 3 ; 7 ) = AP ( a ; d ) , then a + d equals ______. [2019]

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 $AP (1; 3) AP (2; 5) AP (3; 7) = AP (a; d)$ , then $a + d$ equals ______. [2019]

Smart Achievers · Accepted Answer

(157)

$AP (1, 3) : 1, 4, 7, 10, 13, \dots$

$AP (2, 5) : 2, 7, 12, 17, 22, \dots$

$AP (3, 7) : 3, 10, 17, 24, 31, \dots$

$For AP (1, 3) \cap AP (2, 5) \cap AP (3, 7)$ first term will be the minimum common value of a term.

$∴$ we need to find that minimum number which.

when divided by 7 leaves remainder $3 \to (7 m + 3)$

and when divided by 5 leaves remainder $2 \to (5 p + 2)$

and when divided by 3 leaves remainder $1 \to (3 q + 1)$

By hit and trial 52 is such number $(7 \times 7 + 3)$

$∴$ first term $' a'$ of intersection AP = 52

Also common difference $' d'$ of intersection AP

$= LCM (7, 5, 3) = 105$

$∴$ $a + d = 52 + 105 = 157$

Solution

Q.
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 $AP (1; 3) AP (2; 5) AP (3; 7) = AP (a; d)$ , then $a + d$ equals ______. [2019]

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Solution

Q. 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(1;3)AP(2;5)AP(3;7)=AP(a;d), then a+d equals ______. [2019]

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Q.
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 $AP (1; 3) AP (2; 5) AP (3; 7) = AP (a; d)$ , then $a + d$ equals ______. [2019]