Question

If $2 p , p$ and $\left[ p ^2-14\right], p \in R -\{0\}$ are the first three terms of a G.P. in order, then find the $50^{\text {th }}$ term of the sequence, $p, 3 p, 6 p, 10 p, \ldots \ldots$. .
[Note: $[y]$ denotes greatest integer function of $y$.

Answer 1

$p^2=2 p\left[p^2-14\right]$
$p =2\left( p ^2-14\right) \Rightarrow 2 p ^2- p -28=0 $
$\Rightarrow p =4, \frac{-7}{2}$
$\Rightarrow p =4$
$p , 3 p , 6 p , 10 p , \ldots . . $
$T _{ n }=\frac{ n ( n +1)}{2} \cdot p $
$\therefore T _{50}=\frac{50 \times 51}{2} \times 4 \Rightarrow 5100$