SATHEE: Sequences and Series 3 Question 16

Sequences and Series 3 Question 16

16. If the $m$ th, $n$ th and $p$ th terms of an AP and GP are equal and are $x, y, z$ , then prove that $x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = 1$ .

(1979, 3M)

Show Answer

Solution:

Let $a, d$ be the first term and common difference of an $A P$ and $b, r$ be the first term and common ratio of a GP. Then,

$\begin{aligned} x = a + (m - 1) d and x = b r^{m - 1} \\ y = a + (n - 1) d and y = b r^{n - 1} \\ z = a + (p - 1) d and z = b r^{p - 1} \end{aligned}$

Now, $x - y = (m - n) d, y - z = (n - p) d$

and $z - x = (p - m) d$

Again now, $x^{y - z} \cdot y^{z - x} \cdot z^{x - y}$

$\begin{aligned} = {[b r^{m - 1}]}^{(n - p) d} \cdot {[b r^{n - 1}]}^{(p - m) d} \cdot {[b r^{p - 1}]}^{(m - n) d} \\ = b^{[n - p + p - m + m - n] d} \cdot r^{[(m - 1) (n - p) + (n - 1) (p - m) + (p - 1) (m - n)] d} \\ = b^{0} \cdot r^{0} = 1 \end{aligned}$

Play Store

App Store

Ask SATHEE

Welcome to SATHEE !
Select from 'Menu' to explore our services, or ask SATHEE to get started. Let's embark on this journey of growth together! 🌐📚🚀🎓

I'm relatively new and can sometimes make mistakes.
If you notice any error, such as an incorrect solution, please use the thumbs down icon to aid my learning.
To begin your journey now, click on "I understand".

Sequences and Series 3 Question 16

16. If the m th, nth and p th terms of an AP and GP are equal and are x,y,z, then prove that xy−z⋅yz−x⋅zx−y=1.

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

16. If the $m$ th, $n$ th and $p$ th terms of an AP and GP are equal and are $x, y, z$ , then prove that $x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = 1$ .