Question

In the sequences $2,5,8,\dots$ upto $50$ terms and $3,5,7,\dots$ upto $60$ terms, find the number of identical terms.

Answer 1

$\begin{array}{ccc}A{P}^{Ist}\to 2,5,8,......{.50}^{th}term\Rightarrow a=2,d=3,& & \\ A{P}^{2nd}\to 3,5,7,9,....{.60}^{th}term\Rightarrow a=3,d=2& & \\ Forgeneralterm{I}^{st}A.P.\Rightarrow \begin{array}{c}{a}_{n_1}=2+\left(n_1-1\right)3\\ a_{n_1}=3n_1-1\end{array}& & \\ Forgeneralterm{2}^{nd}A.P\Rightarrow \begin{array}{c}{a}_{n_2}=3+\left(n_2-1\right)2\\ a_{n_2}=2n_2+1\end{array}& & \\ If{a}_{n_1}=a_{n_2}& & \\ 3n_1-1=2n_2+1\Rightarrow 3n_1-2n_2=2& & \\ \Rightarrow 3n_1=2+2n_2\Rightarrow 3n_1=2\left(n_2+1\right)\Rightarrow \left[\frac{n_1}{2}+\frac{n_2+1}{3}=k\right]& & \\ \frac{n_1}{2}=k& \frac{n_2+1}{3}=k& \\ n_1=2k& n_2=3k-1& \\ n_1\le 50& n_2\le 60& \left[Givenabove\right]\\ 2k\le 50& 3k\le 61& \\ k\le 25& k\le \frac{61}{2}\approx \mathrm{20.3....}& \\ k\le 20Sopossiblevalueofk=1,2,3,.....20& & \end{array}$

There are 20 identical terms in A.P.