Question

Find the 4 numbers in ap.whose sum is 50 and in which the greatest number is 4 the smallest

Answer 1

Correct Question:

Find the four numbers in A.P. Whose sum is 50 and in which the greatest number is 4 times the smallest number

$\huge\mathfrak\blue{Answer:}$

Arithmetic Progression:

• A sequence of number that increase or decrease by same amount each time
• For ex : 2 , 4 , 6 , 8 ...

Given:

• We have been given 4 numbers in A.P whose sum is 50
• The greatest number is 4 times the smallest number

To Find:

• We have to find all 4 numbers in A.P

Solution:

Let the 4 numbers in A.P. are as follows :

A - 3D , A - D , A + D , A + 3D

$\underline{\large\mathfrak\red{According \: to \: the \: Question }}$

$\boxed{\sf{Sum \: of \: numbers = 50} }$

$\sf{ A - 3D + A - D + A + D + A + 3D = 50}$

$\sf{ 4A = 50}$

$\boxed{\sf{A = \dfrac{50}{4}}}$

_______________________________

$\underline{\large\mathfrak\red{According \: to \: the \: Question }}$

$\boxed{\sf{Greatest \: Number = 4 \: ( Smallest \: Number )} }$

$\implies \sf{A + 3D = 4 \: ( A - 3D)}$

$\implies \sf{A + 3D = 4A - 12D}$

$\implies \sf{3A = 15D}$

$\implies \sf{D = \dfrac{A}{5}}$

$\implies \sf{D = \dfrac{1}{5} \times \dfrac{50}{4}}$

$\implies \boxed{\sf{D = \dfrac{10}{4}} }$

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Substituting and finding Numbers in A.P

$\sf{A - 3D = \dfrac{50}{4} - 3 \left ( \dfrac{10}{4} \right ) = \dfrac{50}{4} - \dfrac{30}{4}}= 5$

$\sf{ }$

$\sf{A - D = \dfrac{50}{4} - \dfrac{10}{4} = \dfrac{40}{4} = 10}$

$\sf{ }$

$\sf{A + D = \dfrac{50}{4} + \dfrac{10}{4} = \dfrac{60}{4} = 15}$

$\sf{ }$

$\sf{A + 3D = \dfrac{50}{4} + 3 \left ( \dfrac{10}{4} \right ) = \dfrac{50}{4} + \dfrac{30}{4}}= 20$

______________________________

$\huge\underline{\sf{\red{A}\orange{n}\green{s}\pink{w}\blue{e}\purple{r}}}$

$\large\boxed{\sf{Required \: Numbers = 5, 10, 15, 20}}$

______________________________

$\huge\mathtt\green{Verification:}$

☞ Sum of Numbers in AP is 50

$\mapsto \sf{Sum = 5 + 10 + 15 + 20}$

$\mapsto \boxed{\sf{Sum = 50}}$

☞ Greatest Number is 4 times the Smallest

$\mapsto \sf{4 \times Smallest \: Number }$

$\mapsto \sf{4 \times 5}$

$\mapsto \sf{20 = Greatest \: Number }$

$\large\red{\underline{\underline{\sf{Hence \: Verified \: !!! }}}}$