1. Tardigrade
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  3. Mathematics
  4. Let ' p ' be the greatest integer for which 5 m 2-16,2 pm , p 2 are distinct consecutive terms of an A.P. where m ∈ R. If common difference of the A.P. is ((A/B)), A, B ∈ N then find the least value of (26 B - A )

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Q. Let ' $p$ ' be the greatest integer for which $5 m ^2-16,2 pm , p ^2$ are distinct consecutive terms of an A.P. where $m \in R$. If common difference of the A.P. is $\left(\frac{A}{B}\right), A, B \in N$ then find the least value of $(26 B - A )$

Sequences and Series

Solution:

$5 m^2-16+p^2=4 p m$
$5 m ^2-4 pm + p ^2-16=0 $
$D \geq 0 \quad p \in[-\sqrt{80}, \sqrt{80}]$
$\therefore$ Greatest integral value of ' $p$ ' is 8 .
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$5 m ^2-16,2 pm , p ^2 $
$\Rightarrow \frac{64}{5}, \frac{192}{5}, \frac{320}{5} $
$\therefore \text { Common difference }=\frac{128}{5}$