# Common first term and difference in 3 arithmetic series

Let $$AP(a,d)$$ denote the set of all the term of an infinite arithmetic progression with first term a and common difference d>0. If $$AP(1,3)\ \cap AP(2,5)\ \cap AP(3,7)=AP(a,d)$$ then find the value of a+d.

My approach

$$1 + \left( {{n_1} - 1} \right)3 = X$$

$$2 + \left( {{n_2} - 1} \right)5 = X$$

$$3 + \left( {{n_3} - 1} \right)7 = X$$

$$1 + \left( {{n_1} - 1} \right)3 = 2 + \left( {{n_2} - 1} \right)5 = 3 + \left( {{n_3} - 1} \right)7 \Rightarrow 3{n_1} - 2 = 5{n_2} - 3 = 7{n_3} - 4 = X$$

$${n_1} = \frac{{X + 2}}{3};{n_2} = \frac{{X + 3}}{5};{n_3} = \frac{{X + 4}}{7}$$

We need to find the value where $$n_1, n_2,n_3$$ are integer and through excel I found the first term as 52.

How do we calculate the first term a=52, d=LCM(3,5,7)=105

• I'm not sure what you're question is. Your working seems correct to me. $a = 52$ and $d = 105$, so $a+ d = 157.$ What's the issue? Dec 18, 2020 at 19:26
• @AdamRubinson I’m guessing they want to know how to find the first term without using excel. Dec 18, 2020 at 19:35
• Not they but I , The formula is correct I entered it into excel and then found the correct answer but the same need to be solved mathematically Dec 18, 2020 at 19:40
• I'm not sure what you mean when you say, "the same need to be solved mathematically". Dec 18, 2020 at 19:42
• @SamarImamZaidi ‘They’ is used when referring to another person. For your problem, using the fact that each of those three expressions must be an integer might help. Dec 18, 2020 at 19:46

$$n_1,n_2,n_3$$ must be integers, which only happens when $$X=3p+1=5q+2=7r+3$$ for $$p,q,r \in \mathbb Z$$. This means $$3p=5q+1=7r+2$$ The easiest way to proceed from here is to consider $$7r+2$$,it needs to be a multiple of $$3$$ and one more than a multiple of $$5$$. Since $$7r+2 \equiv r+2 \equiv 0 \pmod 3$$, you need $$r\equiv 1 \pmod 3$$. Then it is quick and easy to check $$r=1,4,7$$ and $$7$$ is the smallest $$r$$ that meets all the conditions. So, $$a=X=7(7)+3 =52$$