Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
12.3k views
in Arithmetic Progression by (55.4k points)
closed by

Divide 20 into four parts which are in A.P. such that the ratio of the product of the first and fourth is to the product of the second and third is 2 : 3.

1 Answer

+1 vote
by (44.5k points)
selected by
 
Best answer

Let the four parts which are in AP are

(a – 3d), (a – d), (a + d), (a + 3d)

According to question,

The sum of these four parts = 20

⇒(a – 3d) + (a – d) + (a + d) + (a + 3d) = 20

⇒ 4a = 20

⇒ a = 5 …(i)

Now, it is also given that

product of the first and fourth : product of the second and third = 2 : 3

i.e. (a – 3d) × (a + 3d) : (a – d) × (a + d) = 2 : 3

⇒ 3(a2 – 9d2) = 2(a2 – d2)

⇒ 3a2 – 27d2 = 2a2 – 2d2

⇒ 3a2 – 2a2 = – 2d2 + 27d2

⇒ (5)2 = – 2d2 + 27d2 [from (i)]

⇒ 25 = 25d2

⇒ 1 = d2

⇒ d = ±1

Case I: if d = 1 and a = 5

a – 3d = 5 – 3(1) = 5 – 3 = 2

a – d = 5 – 1 = 4

a + d = 5 + 1 = 6

a + 3d = 5 + 3(1) = 5 + 3 = 8

Hence, the four parts are

2, 4, 6, 8

Case II: if d = – 1 and a = 5

a – 3d = 5 – 3( – 1) = 5 + 3 = 8

a – d = 5 – ( – 1) = 5 + 1 = 6

a + d = 5 + ( – 1) = 5 – 1 = 4

a + 3d = 5 + 3( – 1) = 5 – 3 = 2

Hence, the four parts are

8, 4, 6, 2

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...