Question

The number of terms common to the series $3+7+11+\cdots +2019$ and $1+6+11+\cdots +2021$ is

Answer 1

For first series,
$a_1=3,d_1=4$
For second series,
$a_2=1,d_2=5$
For both series, first common term $=11$
second common term = $11+\text{LCM}(4,5)=31$
So, common term sequence is $11,31,51,\dots$
$\because$ last term $\le 2019$
$\Rightarrow 11+(n-1)20\le 2019$
$\Rightarrow n\le 101.4$
Hence, number of common terms is, $n=101$