Question:

Step 1: To find the least value of $K$ for which $4^{1+x}+4^{1-x}$, $\dfrac{K}{2}$, $16^x+16^{-x}$ are three consecutive terms of an arithmetic progression (A.P.), we should start by understanding what it means for terms to be in an A.P. and how we can use this information to solve for $K$. In an A.P., the difference between consecutive terms is constant. This gives us a relationship that we can use to find $K$. Step 2: We are given that the terms $4^{1+x}+4^{1-x}$, $\dfrac{K}{2}$, and $16^x+16^{-x}$ are in an A.P. We can express the relationship between these terms using the definition of an A.P.: the difference between the second term and the first term is equal to the difference between the third term and the second term. This relationship can be written as: $\dfrac{K}{2} - (4^{1+x}+4^{1-x}) = (16^x+16^{-x}) - \dfrac{K}{2}$. Step 3: To simplify the equation and solve for $K$, let's first simplify the expressions $4^{1+x}+4^{1-x}$ and $16^x+16^{-x}$. $4^{1+x}+4^{1-x}$ can be rewritten as $4 \cdot 4^x + 4 \cdot 4^{-x}$, which is $4 \cdot 4^x + \dfrac{4}{4^x}$. $16^x+16^{-x}$ can be rewritten as $(4^2)^x + (4^2)^{-x}$, which is $4^{2x} + \dfrac{1}{4^{2x}}$ or $(4^x)^2 + \left(\dfrac{1}{4^x}\right)^2$. Step 4: Now, let's consider a specific case to find the least value of $K$. If we set $x = 0$, we can simplify our expressions significantly because $4^0 = 1$ and $16^0 = 1$. When $x = 0$, $4^{1+x}+4^{1-x}$ becomes $4^{1+0}+4^{1-0} = 4 + 4 = 8$. When $x = 0$, $16^x+16^{-x}$ becomes $16^0 + 16^0 = 1 + 1 = 2$. Substituting these values into our equation, we get $\dfrac{K}{2} - 8 = 2 - \dfrac{K}{2}$. Step 5: Solving the equation $\dfrac{K}{2} - 8 = 2 - \dfrac{K}{2}$ for $K$ will give us the least value of $K$. First, we add $\dfrac{K}{2}$ to both sides to get $\dfrac{K}{2} + \dfrac{K}{2} - 8 = 2$. This simplifies to $K - 8 = 2$. Then, we add $8$ to both sides to solve for $K$: $K = 2 + 8$. Thus, $K = 10$. Step 6: From the calculation, we find that the least value of $K$ for which the given terms are in an A.P. is $10$. This corresponds to Option 3. The final answer is: the correct option is Option 3, which is $10$.