Finding the Sum of the Given Series

The question asks us to find the sum of the series:

$3 + 3^2 + 3^3 + \dots + 3^8$

Let's first identify the type of series we are dealing with. We can look at the relationship between consecutive terms.

• The first term is $3$.
• The second term is $3^2 = 9$.
• The third term is $3^3 = 27$.

If we divide the second term by the first term, we get $\frac{9}{3} = 3$. If we divide the third term by the second term, we get $\frac{27}{9} = 3$. Since the ratio between consecutive terms is constant, this is a geometric series or geometric progression.

In a geometric series, we need to identify three key parameters:

• The first term ($a$)
• The common ratio ($r$)
• The number of terms ($n$)

For the given series $3 + 3^2 + 3^3 + \dots + 3^8$:

• The first term, $a$, is the first number in the series, which is $3$.
• The common ratio, $r$, is the constant factor by which each term is multiplied to get the next term. We found this to be $3$.
• The number of terms, $n$, is the total count of terms in the series. The exponents go from $1$ ($3^1=3$) up to $8$ ($3^8$). So there are $8$ terms.

Thus, we have:

• $a = 3$
• $r = 3$
• $n = 8$

The formula for the sum ($S_n$) of the first $n$ terms of a geometric series is:

$S_n = a \frac{r^n - 1}{r - 1}$, where $r \neq 1$.

Since our common ratio $r = 3$ is not equal to $1$, we can use this formula to find the sum of the series. We need to find $S_8$.

Calculating the Sum of the Geometric Series

Let's plug in the values of $a$, $r$, and $n$ into the formula:

$S_8 = 3 \times \frac{3^8 - 1}{3 - 1}$

First, we need to calculate $3^8$.

$3^1 = 3$ $3^2 = 9$ $3^3 = 27$ $3^4 = 81$ $3^5 = 243$ $3^6 = 729$ $3^7 = 2187$ $3^8 = 6561$

Now substitute the value of $3^8$ into the formula:

$S_8 = 3 \times \frac{6561 - 1}{3 - 1}$

Simplify the expression inside the fraction:

$S_8 = 3 \times \frac{6560}{2}$

Now perform the division:

$S_8 = 3 \times 3280$

Finally, perform the multiplication:

$S_8 = 9840$

So, the sum of the series $3 + 3^2 + 3^3 + \dots + 3^8$ is $9840$.

This calculation shows how to find the sum of a geometric series using the first term, common ratio, and number of terms.

Revision Table: Key Concepts for Geometric Series Sum

Additional Information on Series Summation

Understanding how to find the sum of different types of series is a fundamental concept in mathematics. Geometric series are common, but there are other types as well.

• Arithmetic Series: In an arithmetic series, the difference between consecutive terms is constant. This constant difference is called the common difference. The sum of an arithmetic series can be found using formulas involving the first term, last term, common difference, or number of terms.
• Infinite Geometric Series: If the absolute value of the common ratio ($|r|$) in a geometric series is less than 1 ($|r| < 1$), the series converges to a finite sum even if it has infinitely many terms. The formula for the sum to infinity is $S_\infty = \frac{a}{1 - r}$. If $|r| \ge 1$ (and $r \neq 1$), the infinite geometric series diverges and does not have a finite sum.
• Other Series: Beyond arithmetic and geometric series, there are many other types, such as harmonic series, power series, and Taylor series, each with its own properties and methods for determining convergence and summation.

Identifying the type of series is the first crucial step in finding its sum. For a geometric series like the one in this problem, identifying the first term, common ratio, and number of terms allows for direct application of the sum formula.