Question

If the nth terms of the two APs: 9, 7, 5, ... and 24, 21, 18,... are the same, find the value of n.

Answer 1

Clearly $9,7,5,...$ is an AP with first term $a=9$ and common difference $d=-2$ and $24,21,18,....$ is an AP with first term $A=24$ and common difference $D=-3$.
Let$a_{n}and{A}_n$ be the $n^{th}$ term of first and second AP respectively.
Therefore according to question,
$a_n=A_n$
$\Rightarrow a+(n-1)d=A+(n-1)D$
$\Rightarrow 9+(n-1)\times -2=24+(n-1)\times -3$
$\Rightarrow -2n+11=-3n+27$
$\Rightarrow n=16$