If n-3- n- and n-1- are in G-P- then n- 5- n- n-1- are in

The correct option is A A.PGiven: n!, (3× n!), (n+1)! are in G.P. ∴ (3× n!)^2=n! × (n+1)! ⇒ 9=n+1 ∴ n=8 Series: n!, (5× n!), (n+1)! ...

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Standard XII

Mathematics

Geometric Progression

If n!,3× n!...

Question

If $n!, 3 \times n!$ and $(n + 1)!$ are in G.P, then $n!, 5 \times n!, (n + 1)!$ are in

A.P

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G.P

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H.P

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None of these

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Solution

The correct option is A A.P
Given: $n!, (3 \times n!), (n + 1)!$ are in G.P.
$∴ (3 \times n!)^{2} = n! \times (n + 1)!$
$\Rightarrow 9 = n + 1$
$∴ n = 8$
Series: $n!, (5 \times n!), (n + 1)!$ becomes $8!, 5 \times 8!, 9!$
For A.P.:
$2 b = a + c$
L.H.S: $10 \times 8!$
R.H.S: $8! + 9! = 8! (1 + 9) = 10 \times 8!$
$∴ L . H . S = R . H . S$

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Similar questions

Q. If

n!

3 (n!)

and

(n + 1)!

are in G.P., then

n!

5 (n!)

and

(n + 1)!

are in

Q. If

n!

3 \times n!

and

(n + 1)!

are in GP then

n!

5 \times n!

and

(n + 1)!

are in

If $S_{1}, S_{2}, . . . . S_{n}$ are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ...., n respectively, then prove that. $S_{1} + S_{2} + 2 S_{3} + 3 S_{4} + . . . . (n - 1)$

$S_{n} = 1^{n} + 2^{n} + 3^{n} + . . . + n^{n}$ .

Q. Assertion :

Let $\displaystyle \Delta _{n}=\begin{vmatrix}

a_{n} & a_{n+3} &a_{n+6} \\

a_{n+1} & a_{n+4} &a_{n+7} \\

a_{n+2} & a_{n+5} & a_{n+8}

\end{vmatrix}$

a_{k} > 0 \forall k \geq 1 a n d a_{1}, a_{2}, a_{3} . . .

are in G.P then

Δ_{n} = 0 \forall n \geq 1

Reason: If

a_{1}, a_{2}, a_{3} . . .

are in A.P then

Δ_{n} = 0 \forall n \geq 1

If the sums of n terms of two arithmetic progressions are in the ratio 3n+5:5n-7, then their n^th terms are in the ratio
(a) 3n-15:n-1
(b) 3n+15:n+1
(c) 5n+13:n+1
(d) 5n-13:n-1

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If n!,3×n! and (n+1)! are in G.P, then n!,5×n!,(n+1)! are in

The correct option is A A.PGiven: n!,(3×n!),(n+1)! are in G.P.∴(3×n!)2=n!×(n+1)!⇒9=n+1∴n=8Series: n!,(5×n!),(n+1)! becomes 8!,5×8!,9!For A.P.:2b=a+cL.H.S: 10×8!R.H.S: 8!+9!=8!(1+9)=10×8!∴L.H.S=R.H.S

If $n!, 3 \times n!$ and $(n + 1)!$ are in G.P, then $n!, 5 \times n!, (n + 1)!$ are in