The sum of first $20$ terms of the sequence $0.7,0.77,0.777, . . . $ is
The sum of first $20$ terms of the sequence $0.7,0.77,0.777, . . . $ is
- [JEE MAIN 2013]
- A
$\frac{7}{{18}}\left( {179 - {{10}^{ - 20}}} \right)$
- B
$\;\frac{7}{9}\left( {99 - {{10}^{ - 20}}} \right)$
- C
$\;\frac{7}{{81}}\left( {179 + {{10}^{ - 20}}} \right)$
- D
$\;\frac{7}{9}\left( {99 + {{10}^{ - 20}}} \right)$
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What will $Rs.$ $500$ amounts to in $10$ years after its deposit in a bank which pays annual interest rate of $10 \%$ compounded annually?
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Suppose four distinct positive numbers $a_1, a_2, a_3, a_4$ are in $G.P.$ Let $b_1=a_1, b_2=b_1+a_2, b_3=b_2+a_3$ and $b_4=b_3+a_4$.
$STATEMENT-1$ : The numbers $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4$ are neither in $A.P$. nor in $G.P.$ and
$STATEMENT-2$ : The numbers $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4$ are in $H.P.$
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$STATEMENT-1$ : The numbers $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4$ are neither in $A.P$. nor in $G.P.$ and
$STATEMENT-2$ : The numbers $\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4$ are in $H.P.$
- [IIT 2008]
If $\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^{9}}+\ldots . .+\frac{10240}{3}=2^{ n } \cdot m$, where $m$ is odd, then $m . n$ is equal to
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- [JEE MAIN 2022]
If the ${p^{th}}$,${q^{th}}$ and ${r^{th}}$ term of a $G.P.$ are $a,\;b,\;c$ respectively, then ${a^{q - r}}{b^{r - p}}{c^{p - q}}$ is equal to
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