Download App

STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
Question
Given $a_1,a_2,a_3.....$ form an increasing geometric progression with common ratio $r$ such that $log_8a_1 + log_8a_2 +.....+ log_8a_{12} = 2014,$ then the number of ordered pairs of integers $(a_1, r)$ is equal to
✓

Answer

c
$\log _{8}\left(a_{1} a_{2} \dots a_{12}\right)=2014 \Rightarrow a_{1}^{12}. r^{66}=2^{6042} \dots(i)$

Let $a_{1}=2^{m}$ and $r=2^{n}$

$\therefore(1) \Rightarrow 2^{12 m+66 n}=2^{6042}$     ....$(i)$

$\Rightarrow 2 \mathrm{m}+11 \mathrm{n}=1007 \Rightarrow \mathrm{m}=\frac{1007-11 \mathrm{n}}{2} \in \mathrm{N}$

$\therefore$ allowed values of $n$ are $n=1,3,5,7, \ldots 91$

$\therefore$ total ordered pairs $=46$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If function $f(x) = \left\{ {\begin{array}{*{20}{c}}{x\,\,\,\,\,,}&{{\rm{if}}\,x\,{\rm{\,\,is\,\, rational\,\,}}}\\{1 - x,}&{{\rm{if}}\,x\,{\rm{\,is\,\, irrational\,}}}\end{array},} \right.$ then $f(x)$ is continuous at ...... number of points
→
If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is
→
Number of solutions of the equation $2^x + x = 2^{sin \ x} +  \sin x$ in $[0,10\pi ]$ is -
→
The expression $[x + (x^3-1)^{1/2}]^5 + [x - (x^3-1)^{1/2}]^5$ is a polynomial of degree :
→
Let a line perpendicular to the line 2x – y = 10 touch the parabola $y^2=4(x-9)$ at the point $P.$ The distance of the point $P$ from the centre of the circle$x^2+y^2-14 x-8 y+56=0$ is $.....$
→
If $\int_{0}^{\pi}\left(\sin ^{3} x\right) e^{-\sin ^{2} x} d x=\alpha-\frac{\beta}{e} \int_{0}^{1} \sqrt{t} e^{t} d t$, then $\alpha+\beta$ is equal to $....$
→
If $a_1, a_2, a_3 …………$ an are in $A.P$ and $a_1 + a_4 + a_7 + …………… + a_{16} = 114$, then $a_1 + a_6 + a_{11} + a_{16}$ is equal to
→
Let $z$ be those complex numbers which satisfy $|z+5| \leq 4$ and $z(1+i)+\bar{z}(1-i) \geq-10, i=\sqrt{-1}$ If the maximum value of $Iz +\left.1\right|^{2}$ is $\alpha+\beta \sqrt{2}$, then the value of $(\alpha+\beta)$ is ...... .
→
Let $f:(-2,2) \rightarrow$ IR be defined by$f(x)=\left\{\begin{array}{cl}x[x] & ,-2 < x < 0 \\ (x-1)[x] & , 0 \leq x < 2\end{array}\right.$
Where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y =|f(x)|$ is not continuous and not differentiable, then $m + n$ is equal to $...........$.
→
If $S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+\ldots .$, then $160 \mathrm{~S}$ is equal to....... .
→