The sequence in your image starts as:

$1.22 + 12.2 + 122 + \cdots + 12,200,000$

Part (a): General Formula Using Summation Notation

This appears to be a geometric sequence where each term increases by a factor of 10. The first term ($a$) is 1.22, and the common ratio ($r$) is 10.

The general formula for the $n$-th term of a geometric sequence is:

$a_n = a \cdot r^{n-1}$

Here:

• $a = 1.22$
• $r = 10$

The general formula for the sum of the first $n$ terms using summation notation would be:

$S_n = \sum_{n=1}^{N} 1.22 \cdot 10^{n-1}$

Where $N$ is the number of terms in the sequence. We will need to calculate the number of terms that gives the final term as 12,200,000.

Part (b): Finding the Sum

The sum $S_n$ of a geometric series can be calculated using the formula:

S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{(if r \neq 1)}

Substituting the values:

• $a = 1.22$
• $r = 10$

We need to calculate $n$ such that the last term $1.22 \times 10^{n-1} = 12,200,000$.

I'll calculate this next.The sequence has 8 terms.

Now, to find the sum of the sequence, we can apply the formula for the sum of a geometric series:

$S_n = \frac{a(1 - r^n)}{1 - r}$

Substituting the known values:

• $a = 1.22$
• $r = 10$
• $n = 8$

I'll calculate the sum next.The sum of the sequence is approximately:

$S_8 = 13,555,555.42$

Summary:

• General formula (summation notation): $S_n = \sum_{n=1}^{8} 1.22 \cdot 10^{n-1}$
• Sum of the sequence: $13,555,555.42$

Would you like any further explanation or details on this problem?

Related Questions:

1. What is the formula for the sum of an arithmetic sequence?
2. How do you derive the formula for the sum of a geometric series?
3. Can this geometric series converge if extended indefinitely?
4. What would happen if the common ratio were less than 1?
5. How does the number of terms affect the total sum of a geometric sequence?

Tip:

When working with geometric series, always check if the common ratio is greater than or less than 1, as it will affect whether the series grows or converges.