Question

The number of terms common to the A.P.’s $3,7,11,\dots 407$ and $2,9,16,\dots 709$ is

Answer 1

Find the total number of common terms.

We have given,

$1^{st}$A.P. $3,7,11,15,19,23,27,31,35,39,43,47,51,55......407$

$2^{nd}$A.P $2,9,16,23,30,37,44,51,58,.....709$

Common term$=23,51,....$ which is also in A.P.

Here, $a=23$ and $d=28$

$$\begin{gathered} \therefore T_n\le 407[407\mathrm{is}\mathrm{the}\mathrm{last}\mathrm{term}\mathrm{of}\mathrm{first}\mathrm{series}] \\ \Rightarrow a+(n-1)d\le 407 \\ \Rightarrow 23+(n-1)28\le 407 \\ \Rightarrow 28n-28\le 407-23 \\ \Rightarrow 28n\le 384+28 \\ \Rightarrow 28n\le 412 \\ \Rightarrow n\le \frac{412}{28} \\ \Rightarrow n\le 14.71 \end{gathered}$$

$\because$number of terms can not be in decimal or fraction

$\therefore n=14$

Hence, the number of terms common to the A.P.’s $3,7,11,\dots 407$ and $2,9,16,\dots 709$ is$14$.