If x, y and z are the terms in G.P., then the term x2 + y2, xy + yz, y2 + z2 are in

Option b: GP. Solution: Since x, y, z are in G.P., we can write: y = xr, z = xr2 where r is the common ratio. Let us substitute these into the three terms: 1) T1 = x2 + y2 = x2 + x2 r2 = x2(1+r2) 2) T2 = xy + yz = x(xr) + (xr)(xr2) = x2 r + x2 r3 = x2 r(1+r2) 3) T3 = y2 + z2 = (xr)2 + (xr2)2 = x2 r2 + x2 r4 = x2 r2(1+r2) Observe the ratios: T2/T1 = x2 r(1+r2)/x2(1+r2) = r T3/T2 = x2 r2(1+r2)/x2 r(1+r2) = r Since the common ratio is the same, the three terms are in G.P. Hence, Option B is the correct answer.

If $x, y$ and $z$ are the terms in G.P., then the term $x^2 + y^2, xy + yz, y^2 + z^2$ are in

The correct answer is Option b: GP. Since $x, y, z$ are in G.P., we can write: $$y = xr, \ z = xr^2$$ where $r$ is the common ratio. Let us substitute these into the three terms: 1) $T_1 = x^2 + y^2 = x^2 + x^2 r^2 = x^2(1+r^2)$ 2) $T_2 = xy + yz = x(xr) + (xr)(xr^2) = x^2 r + x^2 r^3 = x^2 r(1+r^2)$ 3) $T_3 = y^2 + z^2 = (xr)^2 + (xr^2)^2 = x^2 r^2 + x^2 r^4 = x^2 r^2(1+r^2)$ Observe the ratios: $$\frac{T_2}{T_1} = \frac{x^2 r(1+r^2)}{x^2(1+r^2)} = r$$ $$\frac{T_3}{T_2} = \frac{x^2 r^2(1+r^2)}{x^2 r(1+r^2)} = r$$ Since the common ratio is the same, the three terms are in G.P. Hence, **Option B** is the correct answer.

Sequence and SeriesMCQMTP March 22Question 1875 of 212

All Questions

If $\displaystyle x, y$ and $\displaystyle z$ are the terms in G.P., then the term $\displaystyle x^2 + y^2, xy + yz, y^2 + z^2$ are in

Options

AAP

BGP

CHP

DNone of these

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option b — GP

All Options:

AAP
BGP
CHP
DNone of these

Detailed Solution & Explanation

Since

\displaystyle x, y, z

are in G.P., we can write:

y = xr, \ z = xr^2

where

\displaystyle r

is the common ratio.

Let us substitute these into the three terms:
1)

\displaystyle T_1 = x^2 + y^2 = x^2 + x^2 r^2 = x^2(1+r^2)

\displaystyle T_2 = xy + yz = x(xr) + (xr)(xr^2) = x^2 r + x^2 r^3 = x^2 r(1+r^2)

\displaystyle T_3 = y^2 + z^2 = (xr)^2 + (xr^2)^2 = x^2 r^2 + x^2 r^4 = x^2 r^2(1+r^2)

Observe the ratios:

\frac{T_2}{T_1} = \frac{x^2 r(1+r^2)}{x^2(1+r^2)} = r

\frac{T_3}{T_2} = \frac{x^2 r^2(1+r^2)}{x^2 r(1+r^2)} = r

Since the common ratio is the same, the three terms are in G.P.
Hence, **Option B** is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

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If x,y\displaystyle x, yx,y and z\displaystyle zz are the terms in G.P., then the term x2+y2,xy+yz,y2+z2\displaystyle x^2 + y^2, xy + yz, y^2 + z^2x2+y2,xy+yz,y2+z2 are in