If 3 k - k - k 2-34- are the first three terms of a geometric progression- where k - R -and -is the greatest integer function- then the value of - r -110 r k - 2 isB- 385C- 3025D- 325

The correct option is C 30253k, k, [k^2 34] are in G.P. ⇒ k^2=3k[k^2 34] ⇒ k=3[k^2 34] ⇒ k nbsp; is an integer ⇒ k^2 34 nbsp; is also an integer ⇒ 3k^2 k 1 ...

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Standard XII

Mathematics

Sigma n3

If 3 k , k ,[...

Question

If $3 k, k, [k^{2} - 34]$ are the first three terms of a geometric progression, where $k \in R^{+}$ and $[.]$ is the greatest integer function, then the value of $10 \sum r = 1 (r)^{k / 2}$ is

55

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385

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3025

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325

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Solution

The correct option is C $3025$
$3 k, k, [k^{2} - 34]$ are in G.P.
$\Rightarrow k^{2} = 3 k [k^{2} - 34]$
$\Rightarrow k = 3 [k^{2} - 34]$
$\Rightarrow k$ is an integer
$\Rightarrow k^{2} - 34$ is also an integer
$\Rightarrow 3 k^{2} - k - 102 = 0 \Rightarrow k = 6 or k = - \frac{17}{3}$
As $k$ is an integer, $k = 6$

Now, $10 \sum r = 1 (r)^{k / 2} = 10 \sum r = 1 (r)^{3}$
$= 1^{3} + 2^{3} + \dots + 10^{3}$
$= {(\frac{10 \times 11}{2})}^{2} = 3025$

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If 3k, k, [k2−34] are the first three terms of a geometric progression, where k∈R+ and [.] is the greatest integer function, then the value of 10∑r=1(r)k/2 is

The correct option is C 30253k, k, [k2−34] are in G.P. ⇒k2=3k[k2−34] ⇒k=3[k2−34] ⇒k is an integer ⇒k2−34 is also an integer ⇒3k2−k−102=0⇒k=6 or k=−173 As k is an integer, k=6 Now, 10∑r=1(r)k/2=10∑r=1(r)3 =13+23+⋯+103 =(10×112)2=3025

If $3 k, k, [k^{2} - 34]$ are the first three terms of a geometric progression, where $k \in R^{+}$ and $[.]$ is the greatest integer function, then the value of $10 \sum r = 1 (r)^{k / 2}$ is