NIMCET 2022 — Mathematics PYQ
NIMCET | Mathematics | 2022If a1,a2,…,an are any real numbers and n is any positive integer, then which of the following is true?
Choose the correct answer:
- A.
n\sum_{i=1}^{n}a_{i}^{2}<\left(\sum_{i=1}^{n}a_{i}\right)^{2}
- B.
n∑i=1nai2≥(∑i=1nai)2
n∑i=1nai2≥(∑i=1nai)2
Explanation
This inequality is a direct application of the Cauchy-Schwarz Inequality.
The Cauchy-Schwarz Inequality states that for any two sequences of real numbers (a1,a2,…,an) and (b1,b2,…,bn):
(i=1∑naibi)2≤(i=1∑nai2)(i=1∑nbi2)
To derive the given expression, let b1=b2=⋯=bn=1. Substituting these into the formula:
(i=1∑nai⋅1)2≤(i=1∑nai2)(i=1∑n12)
Since ∑i=1n12=1+1+⋯+1=n, we get:
(i=1∑nai)2≤(i=1∑nai2)⋅n
Rearranging this, we obtain:
ni=1∑nai2≥(i=1∑nai)2
This proves that the correct inequality is option B.
Correct Option: B
Explanation
This inequality is a direct application of the Cauchy-Schwarz Inequality.
The Cauchy-Schwarz Inequality states that for any two sequences of real numbers (a1,a2,…,an) and (b1,b2,…,bn):
(i=1∑naibi)2≤(i=1∑nai2)(i=1∑nbi2)
To derive the given expression, let b1=b2=⋯=bn=1. Substituting these into the formula:
(i=1∑nai⋅1)2≤(i=1∑nai2)(i=1∑n12)
Since ∑i=1n12=1+1+⋯+1=n, we get:
(i=1∑nai)2≤(i=1∑nai2)⋅n
Rearranging this, we obtain:
ni=1∑nai2≥(i=1∑nai)2
This proves that the correct inequality is option B.
Correct Option: B