At what values of the parameter a are there values of x such that the numbers 51+x+51−x, 2a, 25x+25−x form an arithmetic progression?

51+x+51−x,2a,25x+25−x forms an A.P⟹a=51+x+51−x+52x+5−2xAM≥GM451+x+51−x+52x+5−2x≥451+x+1−x+2x−2x⟹a≥45

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At what values of the parameter a are there values of x such t

Question

At what values of the parameter $a$ are there values of $x$ such that the numbers $5^{1 + x} + 5^{1 - x}, \frac{a}{2}, 2 5^{x} + 2 5^{- x}$ form an arithmetic progression?

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Emma Discussed

At what values of the parameter

a

are there values of

x

such that the numbers

5^{1 + x} + 5^{1 - x}, \frac{a}{2}, 2 5^{x} + 2 5^{- x}

form an arithmetic progression?

15 mins ago

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15 mins ago

5^{1 + x} + 5^{1 - x}, \frac{a}{2}, 2 5^{x} + 2 5^{- x}

forms an

A . P

⟹ a = 5^{1 + x} + 5^{1 - x} + 5^{2 x} + 5^{- 2 x}

A M \geq GM

\frac{5 ^{1 + x} + 5 ^{1 - x} + 5 ^{2 x} + 5 ^{- 2 x}}{4} \geq 4 5^{1 + x + 1 - x + 2 x - 2 x}

⟹ a \geq 45

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Question Text	At what values of the parameter $a$ are there values of $x$ such that the numbers $5^{1 + x} + 5^{1 - x}, \frac{a}{2}, 2 5^{x} + 2 5^{- x}$ form an arithmetic progression?
Answer Type	Text solution:1
Upvotes	150