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 $$ x $$.

Question

UpStudy AI · Accepted Answer

Let's break down the problem step by step to find the value of $$ x $$.

**Given Series:**
$$ 2^{x} + 2 \cdot 2^{x} + 3 \cdot 2^{x} + \ldots $$

**Understanding the Series:**
Each term in the series can be expressed as:
$$ n \cdot 2^{x} $$
where $$ n $$ is the term number (starting from 1).

**Sum of the First 20 Terms:**
The sum $$ S $$ of the first 20 terms is:
$$ S = \sum_{n=1}^{20} n \cdot 2^{x} $$

Factor out $$ 2^{x} $$:
$$ S = 2^{x} \sum_{n=1}^{20} n $$

The sum of the first 20 natural numbers is:
$$ \sum_{n=1}^{20} n = \frac{20 \cdot (20 + 1)}{2} = 210 $$

So,
$$ S = 210 \cdot 2^{x} $$

**Given:**
$$ S = 1680 $$
$$ 210 \cdot 2^{x} = 1680 $$

**Solving for $$ x $$:**
$$
\begin{align*}
2^{x} & = \frac{1680}{210} \
2^{x} & = 8 \
2^{x} & = 2^{3} \
x & = 3
\end{align*}
$$

**Conclusion:**
The value of $$ x $$ is **3**.

**Answer:**  
The value of $$ x $$ is 3.
The value of $$ x $$ is 3.

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 \( x \).