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Solve the equation: ${ 2 }^{ x }+{ 2 }^{ x+1 }+{ 2 }^{ x+2 }+{ 2 }^{ x+3 }=\frac { 15 }{ 2 }$.

 Jan 23, 2021
 #1
avatar+536 
+1

15/2 = 7.5

Adding multiples of 2 we can never get 1/2 unless we use an exponent of -1 at one point.

Having exponents smaller would result in longer decimals(.25,.125) So x has to be -1

Checking we get

2^-1+2^0+2^1+2^2

1/2+1+2+4

1/2+7

7.5

 Jan 23, 2021
 #2
avatar+31703 
+1

Here's another way of looking at it:

 

 Jan 23, 2021
 #3
avatar+267 
+1

just ignore the base and just look at the exponent

 

so the equation becomes (x)+(x+1)+(x+2)+(x+3)=15/2. 

 

combine like terms -> 4x+6=15/2

 

subtract 6 from both sides and divide both sides by 4 to get x=3/8

 Jan 24, 2021
 #4
avatar+266 
+6

\(2^x+2^{x+1}+2^{x+2}+2^{x+3}=\frac{15}{2}\)

\(15\cdot \:2^x=\frac{15}{2} \)
\(\frac{15\cdot \:2^x}{15}=\frac{\frac{15}{2}}{15}\)

\(2^x=\frac{1}{2}\)

\(x=-1\)

 

 

cheekycheekycheeky

.
 Jan 24, 2021

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