Question

Each of two arithmetic progressions 2, 4, 6, ... and 3, 6, 9, ... is taken up to 200 terms. How many terms are common in these two progressions?

Answer 1

AP₁: a₁​= 2, d₁ ​= 2

AP₂: a₂ ​= 3, d₂ ​= 3

Each A.P. is taken up to 200 terms.

nth term of AP₁: T_n​ = 2n

mth term of AP₂: T_m​ = 3m

Common terms must be divisible by the LCM of 2 and 3, i.e., LCM(2, 3) = 6

So common terms are 6, 12, 18, 24,…

200th term of AP₁:

T_200​ = 2 × 200 = 400

200th term of AP₂:

T_200​ = 3 × 200 = 600

So common terms can go up to 400.

Common terms form the A.P. = 6, 12, 18, …, 400

Number of terms = `400/6`

= 66.6

Number of common terms = 66