Question

Find the number of terms common to the two A.P.'s 3, 7, 11, .......407 and 2, 9, 16, ........709

Answer 1

First $A.P^{\prime }s$ sequence is $3,7,11,....,407$

General term will be $4k+3$ $k\le 101$

Second $A.P^{\prime }s$ sequence is $2,9,16,....,709$

General term will be $7p+2$ $p\le 101$

The common terms will be $51,79,...,28m+51$

$28m+51\le 407⟹28k\le 356⟹k\le 12.71$

And adding the $2$ starting number count of the $2$ $A.P^{\prime }s$ i.e. $3$ & $2$

Number of common terms will be $12+2=14$