JEE Main 2023MathematicsSequences and SeriesMediumMCQ

JEE Main 2023 — Sequences and Series Question with Solution

JEE Main 2023 (15 Apr Shift 1)

Question

Let A1 and A2 be two arithmetic means and G1, G2 and G3 be three geometric means of two distinct positive numbers. Then G14+G24+G34+G12G32 is equal to

Choose an option

Show full solutionCorrect option: A
Correct answer
AA1+A22G1G3

Step-by-step explanation

Let the two numbers are a, b

⇒a,A1,A2,b are in AP.

⇒b=a+4-1d

⇒d=b-a3

⇒A1=a+b-a3=2a+b3

⇒A2=a+b-a3·2=a+2b3

Similarly a,G1,G2,G3,b are in GP.

⇒b=ar5-1

⇒r=ba14

⇒G1=aba14

⇒G2=aba24

⇒G3=aba34

=G14+G24+G34+G12·G32

⇒a4·ba+a4·b2a2+a4·b3a3+a4·b2a2

=ba3+b2a2+b3a+a2b2

=aba2+b2+2ab=aba+b2

⇒A1+A22·G1G3=a+b2·ab

Hence this is the correct option.

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About this question

This is a previous-year question from JEE Main 2023, covering the Sequences and Series chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.