Q. 103.7( 49 Votes )

The sum of three numbers in G.P. is 66. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Answer :

Let the three numbers in G.P. be a, ar, and ar2.


According to question -


a + ar + ar2 = 66


a(1 + r + r2) = 66


…(1)


Given that -


(a – 1), (ar – 7), (ar2 – 21) forms an A.P.


thus the common difference between the consecutive terms will be equal.


(ar – 7) – (a – 1) = (ar2 – 21) – (ar – 7)


ar – a – 6 = ar2 – ar – 14


ar2 – 2ar + a = 8


a(r2 – 2r + 1) = 8


…(2)


Comparing equations (1) & (2), we get -



On Cross - multiplying,


66(r2 – 2r + 1) = 8(1 + r + r2)


7(r2 – 2r + 1) = (1 + r + r2)


7r2 – 14r + 7 = 1 + r + r2


6r2 – 16r + 6 = 0


6r2 – 12r - 3r + 6 = 0


6r (r – 2) – 3 (r – 2) = 0


(6r – 3) (r – 2) = 0


r = 2 or r = 1/2


When r = 2, a = 8


When r = 1/2, a = 32


Therefore,


when r = 2, the three numbers in G.P. are 8, 16, and 32.


when r = 1/2, the three numbers in G.P. are 32, 16, and 8.


Thus, in either case, the three required numbers are 8, 16, and 32.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Interactive quiz on inequalities involving AM and GMInteractive quiz on inequalities involving AM and GMInteractive quiz on inequalities involving AM and GM45 mins
Check Your progress Part 2| Interactive Quiz: Sequence & SeriesCheck Your progress Part 2| Interactive Quiz: Sequence & SeriesCheck Your progress Part 2| Interactive Quiz: Sequence & Series47 mins
Sequence & Series (Lecture 1)Sequence & Series (Lecture 1)Sequence & Series (Lecture 1)56 mins
Sequence & Series (Lecture 5)Sequence & Series (Lecture 5)Sequence & Series (Lecture 5)42 mins
Sequence & Series (Lecture 7)Sequence & Series (Lecture 7)Sequence & Series (Lecture 7)37 mins
Use of sigma in finding sumUse of sigma in finding sumUse of sigma in finding sum42 mins
Sequence & Series (Lecture 8)Sequence & Series (Lecture 8)Sequence & Series (Lecture 8)35 mins
Check Your progress Part 3| Interactive Quiz: Sequence & SeriesCheck Your progress Part 3| Interactive Quiz: Sequence & SeriesCheck Your progress Part 3| Interactive Quiz: Sequence & Series46 mins
Questions on relation b/w AM, GM & HMQuestions on relation b/w AM, GM & HMQuestions on relation b/w AM, GM & HM41 mins
Sequence & Series | Quiz ka DangalSequence & Series | Quiz ka DangalSequence & Series | Quiz ka Dangal55 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses