Question

If $a, x, y, b$ are in G.P., then $(x+y)^{2}=$

• A. $(a+x)(y+b)$
• B. $(b+x)(y+a)$
• C. $(a+2x)(y+b)$
• D. $(a+x)(y+2b)$
• E. $(a+2x)(y+2b)$

Hint:

Logic Tip: You can also solve this by picking simple numbers for the G.P. sequence. Let $a=1, x=2, y=4, b=8$ ($r=2$). Then $(x+y)^2 = (2+4)^2 = 36$. Testing Option A: $(1+2)(4+8) = 3 \cdot 12 = 36$.

Answer 1

The Correct Option is A

Concept:
When four terms $a, x, y, b$ are in a Geometric Progression with a common ratio $r$, they can be written in terms of the first term $a$ as: $a, ar, ar^2, ar^3$.
Step 1: Express the variables in terms of a and r.
$x = ar$ $y = ar^2$ $b = ar^3$
Step 2: Expand the given expression $(x+y)^2$.
$$(x+y)^2 = (ar + ar^2)^2$$ Factor out $ar$ from inside the parenthesis: $$= (ar(1 + r))^2$$ $$= a^2r^2(1 + r)^2$$
Step 3: Test the given options to find the match.
Let's expand Option A: $(a+x)(y+b)$ Substitute the variables: $$(a + ar)(ar^2 + ar^3)$$ Factor out $a$ from the first term and $ar^2$ from the second term: $$= a(1 + r) \cdot ar^2(1 + r)$$ Multiply the factors together: $$= a^2r^2(1 + r)^2$$ Since this exactly matches our expanded expression from Step 2, Option A is correct.