The image you uploaded shows a worksheet that appears to ask for finding the sum of the series for each sequence provided. The sequences given are:

1. $2, 4, 6, 8, 10, \ldots$
2. $2, 4, 8, 16, 32, \ldots$
3. $3, 6, 9, 12, 15, \ldots$
4. $1, 2, 4, 8, 16, \ldots$
5. $5, 6, 7, 8, 9, \ldots$

To solve this, you need to identify whether each sequence is arithmetic or geometric, then calculate the sum accordingly.

Let's break down each sequence:

1. Sequence 1: $2, 4, 6, 8, 10, \ldots$

   • Type: Arithmetic (common difference $d = 2$)
   • Sum: $S_n = \frac{n}{2} \times (2a + (n-1)d)$

2. Sequence 2: $2, 4, 8, 16, 32, \ldots$

   • Type: Geometric (common ratio $r = 2$)
   • Sum: $S_n = a \times \frac{1 - r^n}{1 - r}$ for $r > 1$

3. Sequence 3: $3, 6, 9, 12, 15, \ldots$

   • Type: Arithmetic (common difference $d = 3$)
   • Sum: $S_n = \frac{n}{2} \times (2a + (n-1)d)$

4. Sequence 4: $1, 2, 4, 8, 16, \ldots$

   • Type: Geometric (common ratio $r = 2$)
   • Sum: $S_n = a \times \frac{1 - r^n}{1 - r}$ for $r > 1$

5. Sequence 5: $5, 6, 7, 8, 9, \ldots$

   • Type: Arithmetic (common difference $d = 1$)
   • Sum: $S_n = \frac{n}{2} \times (2a + (n-1)d)$

Would you like the sums for a specific number of terms in each series, or do you have another request?

Follow-up Questions:

1. How do you determine whether a sequence is arithmetic or geometric?
2. What is the formula for the sum of an infinite geometric series?
3. How does the common ratio affect the sum of a geometric series?
4. Can you explain the difference between an arithmetic series and an arithmetic sequence?
5. What happens to the sum of a geometric series when the common ratio is less than 1?

Tip:

Remember, identifying the pattern in a sequence (arithmetic or geometric) is crucial for determining the correct formula to use for finding the sum.