Understanding the Given Series

The given series is 6, 13/2, 7, 15/2, ...

Let's write the first few terms clearly:

• First term ($a_1$): 6
• Second term ($a_2$): $\frac{13}{2} = 6.5$
• Third term ($a_3$): 7
• Fourth term ($a_4$): $\frac{15}{2} = 7.5$

Identifying the Type of Series

To determine the type of series, we check the difference between consecutive terms.

• Difference between the second and first term: $a_2 - a_1 = \frac{13}{2} - 6 = \frac{13}{2} - \frac{12}{2} = \frac{1}{2}$
• Difference between the third and second term: $a_3 - a_2 = 7 - \frac{13}{2} = \frac{14}{2} - \frac{13}{2} = \frac{1}{2}$
• Difference between the fourth and third term: $a_4 - a_3 = \frac{15}{2} - 7 = \frac{15}{2} - \frac{14}{2} = \frac{1}{2}$

Since the difference between consecutive terms is constant, the given series is an arithmetic series. The common difference ($d$) is $\frac{1}{2}$.

Sum of an Arithmetic Series Formula

The sum of the first $n$ terms of an arithmetic series is given by the formula:

$$S_n = \frac{n}{2} [2a_1 + (n-1)d]$$

Where:

• $S_n$ is the sum of the first $n$ terms
• $n$ is the number of terms
• $a_1$ is the first term
• $d$ is the common difference

Calculating the Sum of the First 16 Terms

We need to find the sum of the first 16 terms ($n=16$) of the series.

We have:

• $n = 16$
• $a_1 = 6$
• $d = \frac{1}{2}$

Substitute these values into the formula:

$$S_{16} = \frac{16}{2} \left[2(6) + (16-1)\left(\frac{1}{2}\right)\right]$$

Now, let's simplify step-by-step:

$$S_{16} = 8 \left[12 + (15)\left(\frac{1}{2}\right)\right]$$

$$S_{16} = 8 \left[12 + \frac{15}{2}\right]$$

To add 12 and $\frac{15}{2}$, we find a common denominator:

$$12 = \frac{12 \times 2}{2} = \frac{24}{2}$$

So the expression inside the bracket becomes:

$$12 + \frac{15}{2} = \frac{24}{2} + \frac{15}{2} = \frac{24 + 15}{2} = \frac{39}{2}$$

Substitute this back into the equation for $S_{16}$:

$$S_{16} = 8 \left[\frac{39}{2}\right]$

Now, multiply 8 by $\frac{39}{2}$. We can cancel the 2 in the denominator with the 8:

$$S_{16} = \frac{8}{1} \times \frac{39}{2} = \frac{8 \times 39}{2} = 4 \times 39$$

Finally, calculate the product:

$$S_{16} = 4 \times 39 = 156$$

Final Answer

The sum of the first 16 terms of the given series is 156.

Revision Table: Arithmetic Series Sum

Additional Information: Properties of Arithmetic Series

Arithmetic series have several useful properties:

• Any term can be found if the first term and common difference are known.
• The difference between any two terms is directly proportional to the difference in their term numbers. For example, $a_m - a_k = (m-k)d$.
• The sum of terms equidistant from the beginning and end is constant. For instance, in a series of n terms, $a_1 + a_n = a_2 + a_{n-1}$, and so on.
• An arithmetic series can be used to model situations involving linear growth or decay over discrete intervals.