Question

Prove that there are $17$ identical terms in the two A.P.'s $2,5,8,11,$_______$60$ terms and $3,5,7,9$, ________$50$ terms.

Answer 1

Let $T_p=T_q\Rightarrow 3p-1=2q+1$,

Subtract $5$ from both sides

$\therefore 3(p-2)=2(q-2)$

or $\frac{p-2}{2}=\frac{q-2}{3}=k$, say

$\therefore p=2k+2$ and $q=3k+2$

Now p varies from $1$ to $60$ and q varies from $1$ to $50$.

Hence we have the following

$1\le 2k+2\le 60$ and $1\le 3k+2\le 50$

$\therefore -\frac{1}{2}\le k\le 29$ and $-\frac{1}{3}\le k\le 16$.

Clearly $k=0,1,2,3,...,16$ for common values and hence there will be $17$ common terms.