Let a, b, c, d, e be natural numbers in an arithmetic progression such that a+b+c+d+e is cube of an

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8. Sequences and Series

normal

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is

$2$

$3$

$4$

$5$

(KVPY-2013)

Solution

(b)

We have,

$a, b, c, d, e$ are natural number and in AP.

Let $D$ is common difference of $AP$.

$\therefore$ Let $c=C$

$a =C-2 D$
$b =C-D$

$d =C+D$

$e =C+2 D$

$a+b+c+d+e=5 C$

and $b+c+d=3 C$

Given, $a+b+c+d+e$ is a cube of number

$\therefore 5 C=\lambda^3$

and $b+c+d$ is a square of number

$\therefore 3 C=u^2$

From Eqs.$(i)$ and $(ii)$, we get

$\frac{\lambda^3}{5}=\frac{u^2}{3}$

$\lambda^3$ and $u^2$ is a multiple of 15

$\therefore$ Smallest possible value of $\lambda=15$ and $u=45$

$\therefore \quad c=\frac{u^2}{3}=\frac{(45)^2}{3}=675$

$\therefore$ Number of digits $=3$

Standard 11

Mathematics

$8^{th}$ term of the series $2\sqrt 2 + \sqrt 2 + 0 + …..$ will be

easy

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hard

(JEE MAIN-2022)

Let $A B C D$ be a quadrilateral such that there exists a point $E$ inside the quadrilateral satisfying $A E=B E=C E=D E$. Suppose $\angle D A B, \angle A B C, \angle B C D$ is an arithmetic progression. Then the median of the set $\{\angle D A B, \angle A B C, \angle B C D\}$ is

normal

(KVPY-2020)

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normal

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is

Solution

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