Let a, b, c, d, e be natural numbers in an ar | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the arithmetic progression be $a, b, c, d, e$ with common difference $D$. Since $a, b, c, d, e$ are in arithmetic progression,we can write the

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(B) Let the arithmetic progression be $a, b, c, d, e$ with common difference $D$. Since $a, b, c, d, e$ are in arithmetic progression,we can write them as $c-2D, c-D, c, c+D, c+2D$. Given $a+b+c+d+e = 5c = \lambda^3$ for some integer $\lambda$. Given $b+c+d = 3c = u^2$ for some integer $u$. From $3c = u^2$,we have $c = \frac{u^2}{3}$. Substituting this into $5c = \lambda^3$,we get $5(\frac{u^2}{3}) = \lambda^3$,which implies $5u^2 = 3\lambda^3$. For this to hold,$u$ must be a multiple of $3$ and $\lambda$ must be a multiple of $5$. Let $u = 3k$ and $\lambda = 5m$. Then $5(9k^2) = 3(125m^3)$,which simplifies to $45k^2 = 375m^3$,or $3k^2 = 25m^3$. For the smallest natural number solution,we set $m=3$ and $k=5$. Then $u = 3(5) = 15$ and $\lambda = 5(3) = 15$. However,checking the condition $5c = \lambda^3$,if $\lambda=15$,$5c = 3375$,so $c = 675$. Checking $3c = u^2$,$3(675) = 2025 = 45^2$. Thus,$c = 675$,which has $3$ digits.

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 the square of an integer. The least possible value of the number of digits of $c$ is

Similar Questions

If the first term of an $A.P.$ is $10$,the last term is $50$,and the sum of all the terms is $300$,then the number of terms is:

If the sum of $n$ terms of an arithmetic progression is $2n^2 + 5n$,then its $n^{th}$ term is.........

What is the arithmetic mean of the series $a + (a + d) + (a + 2d) + \dots + (a + 2nd)$?

If the sum of the first $n$ terms of an $A.P.$ is $c n^2$,then the sum of the squares of these $n$ terms is

Let there be four distinct integers in an increasing arithmetic progression. One of these integers is equal to the sum of the squares of the other three. What is the product of all these numbers?

Vedclass Test Series

Exam Paper Generator

Online Exam Module