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Global Maxima

If |ak|< 1,...

Question

If $| a_{k} | < 1, λ_{k} \geq 0$ for $k = 1, 2, 3, . . . . n$ and $λ_{1} + λ_{2} + . . . . . λ_{n} = 1$ , then the value of $| λ_{1} a_{1} + λ_{2} a_{2} + . . . . . + λ_{n} a_{n} |$ is

A

Equal to one

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B

Greater than one

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C

Zero

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D

less than one

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Solution

The correct option is D less than one

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Similar questions

1 + λ + λ^{2} + . . . . . + λ^{n} = (1 + λ) (1 + λ^{2}) (1 + λ^{4}) (1 + λ^{8}) (1 + λ^{16})

, then the value of

n

n ϵ N

)

z_{1}, z_{2} - -, z_{n}

are complex numbers such that

| z_{i} | < l a n d λ_{i} > 0

i = 1, 2, - - - n

λ_{1} + λ_{2} + - - + λ_{n} = 1

| λ_{1} z_{1} + λ_{2} z_{2} + - - + λ_{n} z_{1} | ?

a_{k} = \frac{1}{k (k + 1)}, f o r k = 1, 2, 3..... n, t h e n {(n \sum k = 1 a_{k})}_{k}^{2} =

λ_{1} + λ_{2} + . . . + λ_{n} = 1

λ_{i} > 0

i = 1, 2, . . ., n

| a_{i} | < 1

ω

is a complex cube root of unity, then

| λ_{1} a_{1} ω + λ_{2} a_{2} ω^{2} + . . . + λ_{n} a_{n} ω^{n} |

a_{k} = \frac{1}{K (K + 1)}

K = 1, 2, 3, . . . . . n,

{(\sum_{k = 1}^{n} a_{k})}^{2} =

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