2.1.2 Given: \( 2^{x}+2.2^{x}+3.2^{x}+\ldots \) The sum of the first 20 terms of the series is 1680 . Calculate the value of \( x \).

To solve the series sum given by \( S = 2^{x}+2 \cdot 2^{x}+3 \cdot 2^{x}+\ldots + 20 \cdot 2^{x} \), we can factor out \( 2^{x} \): \[ S = 2^{x}(1 + 2 + 3 + \ldots + 20) \] The sum of the first \( n \) natural numbers is given by the formula: \[ \text{Sum} = \frac{n(n+1)}{2} \] For \( n = 20 \): \[ \text{Sum} = \frac{20(20 + 1)}{2} = \frac{20 \cdot 21}{2} = 210 \] Now substituting this back into our equation for \( S \): \[ S = 2^{x} \cdot 210 \] We are given that the sum \( S \) equals 1680: \[ 2^{x} \cdot 210 = 1680 \] Dividing both sides by 210 gives: \[ 2^{x} = \frac{1680}{210} = 8 \] Noticing that \( 8 = 2^3 \), we can equate: \[ 2^{x} = 2^3 \] This implies: \[ x = 3 \] Thus, the value of \( x \) is: \[ \boxed{3} \]