If 2 p , p and [ p 2-14], p ∈ 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.

Question

If 2 p , p and [ p 2-14], p ∈ 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. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

p2=2 p[p2-14] p =2( p 2-14) ⇒ 2 p 2- p -28=0 ⇒ p =4, (-7/2) ⇒ p =4 p , 3 p , 6 p , 10 p , ldots . . T n =( n ( n +1)/2) ⋅ p ∴ T 50=(50 × 51/2) × 4 ⇒ 5100

Q. If $2 p, p$ and $[p^{2} - 14], p \in R - {0}$ are the first three terms of a G.P. in order, then find the $5 0^{th}$ term of the sequence, $p, 3 p, 6 p, 10 p, \dots\dots$ . .
[Note: $[y]$ denotes greatest integer function of $y$ .

$p^{2} = 2 p [p^{2} - 14]$
$p = 2 (p^{2} - 14) \Rightarrow 2 p^{2} - p - 28 = 0$
$\Rightarrow p = 4, \frac{- 7}{2}$
$\Rightarrow p = 4$
$p, 3 p, 6 p, 10 p, \dots ..$
$T_{n} = \frac{n ( n + 1 )}{2} \cdot p$
$∴ T_{50} = \frac{50 \times 51}{2} \times 4 \Rightarrow 5100$

Q. If 2p,p and [p2−14],p∈R−{0} are the first three terms of a G.P. in order, then find the 50th term of the sequence, p,3p,6p,10p,……. . [Note: [y] denotes greatest integer function of y.

p2=2p[p2−14] p=2(p2−14)⇒2p2−p−28=0 ⇒p=4,2−7​ ⇒p=4 p,3p,6p,10p,….. Tn​=2n(n+1)​⋅p ∴T50​=250×51​×4⇒5100

Q. If $2 p, p$ and $[p^{2} - 14], p \in R - {0}$ are the first three terms of a G.P. in order, then find the $5 0^{th}$ term of the sequence, $p, 3 p, 6 p, 10 p, \dots\dots$ . .
[Note: $[y]$ denotes greatest integer function of $y$ .

$p^{2} = 2 p [p^{2} - 14]$
$p = 2 (p^{2} - 14) \Rightarrow 2 p^{2} - p - 28 = 0$
$\Rightarrow p = 4, \frac{- 7}{2}$
$\Rightarrow p = 4$
$p, 3 p, 6 p, 10 p, \dots ..$
$T_{n} = \frac{n ( n + 1 )}{2} \cdot p$
$∴ T_{50} = \frac{50 \times 51}{2} \times 4 \Rightarrow 5100$