(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

If the $4^{\text{th}}$,$10^{\text{th}}$,and $16^{\text{th}}$ terms of a $G.P.$ are $x, y$,and $z$ respectively,prove that $x, y, z$ are in $G.P.$

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In a $G.P.$, if the product of the first three terms is $27$ and the set of all possible values for the sum of its first three terms is $\mathbb{R} - (a, b)$, then $a^{2} + b^{2}$ is equal to . . . . . . .

DifficultJEE MAIN 2026

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Let $A_{1}, A_{2}, A_{3}, \ldots$ be squares such that for each $n \geq 1,$ the length of the side of $A_{n}$ equals the length of the diagonal of $A_{n+1}$. If the side length of $A_{1}$ is $12 \text{ cm}$,then the smallest value of $n$ for which the area of $A_{n}$ is less than $1 \text{ cm}^2$ is:

DifficultJEE MAIN 2021

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The $6^{th}$ term of a $G.P.$ is $32$ and its $8^{th}$ term is $128$,then the common ratio of the $G.P.$ is

Easy

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If the $(p + q)^{th}$ term of a $G.P.$ is $m$ and the $(p - q)^{th}$ term is $n$,then the $p^{th}$ term will be

Medium

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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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If the $4^{\text{th}}$,$10^{\text{th}}$,and $16^{\text{th}}$ terms of a $G.P.$ are $x, y$,and $z$ respectively,prove that $x, y, z$ are in $G.P.$

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