(A) The number of bacteria doubles every hour,forming a geometric progression $(G.P.)$ where the initial number of bacteria $a = 30$ and the common ra

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(A) The number of bacteria doubles every hour,forming a geometric progression $(G.P.)$ where the initial number of bacteria $a = 30$ and the common ratio $r = 2$.The number of bacteria after $t$ hours is given by $a_{t+1} = a \times r^{t} = 30 \times 2^{t}$.At the end of the $2^{\text{nd}}$ hour $(t=2)$: $30 \times 2^{2} = 30 \times 4 = 120$.At the end of the $4^{\text{th}}$ hour $(t=4)$: $30 \times 2^{4} = 30 \times 16 = 480$.At the end of the $n^{\text{th}}$ hour $(t=n)$: $30 \times 2^{n}$.

Class 11 Mathematics Sequences and Series Geometric progression

Medium

The number of bacteria in a certain culture doubles every hour. If there were $30$ bacteria present in the culture originally,how many bacteria will be present at the end of $2^{\text{nd}}$ hour,$4^{\text{th}}$ hour and $n^{\text{th}}$ hour?

A
$120, 480, 30(2)^{n}$
B
$120, 480, 30(2)^{n-1}$
C
$60, 120, 30(2)^{n}$
D
$240, 480, 30(2)^{n+1}$

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Sequences and Series

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Geometric progression

For a sequence $(t_{n})$,if $s_{n} = 7(3^{n} - 1)$,then $t_{n} =$

EasyMHT CET 2020

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If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1 = \frac{1}{8}$ and $a_2 \neq a_1$ is the arithmetic mean of the next two terms and $S_n = a_1 + a_2 + \ldots + a_n$,then $S_{20} - S_{18}$ is equal to

DifficultJEE MAIN 2024

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If $1 + \sin x + \sin^2 x + \dots \text{ to } \infty = 4 + 2\sqrt{3}$,where $0 < x < \pi$,then:

Medium

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Let the positive numbers $a_1, a_2, a_3, a_4$ and $a_5$ be in a $G$.$P$. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively,where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$,then $m+n$ is equal to $.........$.

DifficultJEE MAIN 2023

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Let $a_1, a_2, a_3, \ldots$ be a $GP$ of increasing positive numbers. If the product of the fourth and sixth terms is $9$ and the sum of the fifth and seventh terms is $24$,then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to $.........$.

DifficultJEE MAIN 2023

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The number of bacteria in a certain culture doubles every hour. If there were $30$ bacteria present in the culture originally,how many bacteria will be present at the end of $2^{\text{nd}}$ hour,$4^{\text{th}}$ hour and $n^{\text{th}}$ hour?

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For a sequence $(t_{n})$,if $s_{n} = 7(3^{n} - 1)$,then $t_{n} =$

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