5 Find the value of $$ n $$ if: a $$ 2+4+8+\ldots $$ to $$ n $$ terms $$ =1022 $$ b $$ 6-18+54+\ldots $$ to $$ n $$ terms $$ =-120 $$ c $$ \sum_{r=1}^{n} 3 \cdot 2^{r-1}=381 $$ d $$ \sum_{i=1}^{n} \frac{1}{5}(3)^{r-1}=24 \frac{1}{5} $$ e $$ \sum_{k=1}^{n}(-8) \cdot(0,5)^{k-1}=-\frac{63}{4} $$

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**Problem a**

Find the value of $$ n $$ for the geometric series:
$$ 2 + 4 + 8 + \ldots $$  
given that the sum of $$ n $$ terms is $$ 1022 $$.

**Solution:**

This is a geometric series where:
- First term ($$ a $$) = 2
- Common ratio ($$ r $$) = 2

The sum of the first $$ n $$ terms of a geometric series is given by:
$$ S_n = a \frac{r^n - 1}{r - 1} $$

Substituting the known values:
$$ 1022 = 2 \frac{2^n - 1}{2 - 1} $$
$$ 1022 = 2(2^n - 1) $$
$$ 1022 = 2^{n+1} - 2 $$
$$ 2^{n+1} = 1024 $$

Recognizing that $$ 1024 = 2^{10} $$:
$$ n + 1 = 10 $$
$$ n = 9 $$

**Answer:**  
Problem a Answer:

The value of $$ n $$ is $$ 9 $$.
The value of $$ n $$ is $$ 9 $$.

5 Find the value of if:
a to terms
b to terms
c
d
e

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5 Find the value of if: a to terms b to terms c d e