Question

If A,A-B and A+B are in AP and A,B,A-B and A+B are prime numbers then find A and B give reason​

Answer 1

The AP is taken in the order A - B, A, A + B,... because if we take the AP as A, A - B, A + B,... then value of B will be 0 and hence the AP will be A, A, A,...

We see the common difference of this AP is B.

$\longrightarrow\sf{A-(A-B)=(A+B)-A=B}$

Given that A, B, A - B and A + B all are prime numbers, so they should be odd.

But we know that odd numbers differ each other by an even number.

$\longrightarrow\sf{Odd-Odd=Even}$

This implies the terms in the AP,  A - B, A, A + B,...,  should differ each other so that the common difference of this AP (B) should be even. Hence,

$\longrightarrow\sf{\underline{\underline{B=2}}}$

because B is also a prime number, and 2 is only the even number in the set of prime numbers.

Hence our AP will be A - 2, A, A + 2.

Since the AP is increasing, the first term (A - 2) should be greater than or equal to 3, since it's an odd prime number.

$\longrightarrow\sf{A-2\geq3}$

$\longrightarrow\sf{A\geq5}$

$\longrightarrow\sf{A+2\geq7}$

We can prove that exactly one among the three terms A - 2, A, A + 2 is a multiple of 3, by Euclid's Division Algorithm. But since the three terms are prime numbers, the multiple of 3 among the three terms should be 3.

Since $\sf{A\geq5}$ and $\sf{A+2\geq7,}$ they can't be 3, so $\sf{A-2}$ should be equal to 3.

$\longrightarrow\sf{A-2=3}$

$\longrightarrow\sf{\underline{\underline{A=5}}}$

Hence values of A and B are 5 and 2 respectively, and hence the AP is 3, 5, 7,...