Sequences & Series
AM-GM Relation with Product Expression
nta_pyq_2023_apr
Grade 11
Question:
Let $A_1$ and $A_2$ be two AMs and $G_1,G_2,G_3$ be three GMs of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to
$(A_1+A_2)^2G_1G_3
$2(A_1+A_2)G_1G_3
$(A_1+A_2)G_1^2G_3^2
$2(A_1+A_2)G_1^2G_3^2
Step-by-Step Solution
Key Concept: Numbers $a,b$: $A_1=\frac{2a+b}{3}$, $A_2=\frac{a+2b}{3}$, $A_1+A_2=a+b$. $G_1=ab^{1/4}a^{3/4}$... $G_1G_3=ab$. Expression $=a^3b+a^2b^2+ab^3+a^2b^2=ab(a^2+2ab+b^2)=ab(a+b)^2=(A_1+A_2)^2G_1G_3$.
$(A_1+A_2)^2G_1G_3$.
Correct Answer: 1