Plz answer this question I tried this but not getting the solution
Plz answer this question I tried this but not getting the solution
Priyanka Sharma , 5 Years ago
Grade 11
Analytical Geometry> Plz answer this question I tried this but...
1 Answers
Aditya Gupta
Last Activity: 5 Years ago
hello priyanka. the above ques is a direct application of cauchy schwartz inequality for real numbers (you can google this if you dont know about this inequality).
from cauchy schwartz inequality, we can say that
(x1^2 + x2^2 + …. + x(n-1)^2)*(x2^2 + x3^2 + …. + xn^2) is greater than or equal to (x1x2 + x3x3 + …..... + x(n-1)xn)^2.
however, in the ques we are given the inequality:
(x1^2 + x2^2 + …. + x(n-1)^2)*(x2^2 + x3^2 + …. + xn^2) is less than or equal to (x1x2 + x3x3 + …..... + x(n-1)xn)^2.
so obviously for both ineqs to be true, the given expressions are bound to be equal to each other. hence
(x1^2 + x2^2 + …. + x(n-1)^2)*(x2^2 + x3^2 + …. + xn^2) = (x1x2 + x3x3 + …..... + x(n-1)xn)^2
however, by cauchy ineq., equality holds if and only if ai= kbi for some constant k.
note that here ai= xi and bi= x(i+1)
so that xi= kx(i+1)
or x(i+1)/xi= 1/k = constant (say r).
but this is the very definition of a GP.
hence, x1, x2, …......., xn form a geometric progression.
kindly approve :)
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