Question:
The number of terms common to the two A.P.'s $3,7,11, \ldots$, 407 and $2,9,16, \ldots, 709$ is__________.
Solution:
First common term of both the series is 23 and
common difference is $7 \times 4=28$
$\because \quad$ Last term $\leq 407$
$\Rightarrow \quad 23+(n-1) \times 28 \leq 407$
$\Rightarrow \quad(n-1) \times 28 \leq 384$
$\Rightarrow \quad n \leq \frac{384}{28}+1$
$\Rightarrow n \leq 14.71$
Hence, $n=14$
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