The given sequence is an arithmetic progression with the first term $a = -\frac{5}{4}$ and the common difference $d$.

To find $d$, we subtract the first term from the second term:

$d = -\frac{3}{4} - \left(-\frac{5}{4}\right) = -\frac{3}{4} + \frac{5}{4} = \frac{2}{4} = \frac{1}{2}$

So, the common difference $d = \frac{1}{2}$.

Step 1: Find $S_6$ (the sum of the first 6 terms)

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

$S_n = \frac{n}{2} \left(2a + (n-1) d\right)$

For $n = 6$, we substitute the values:

• $a = -\frac{5}{4}$
• $d = \frac{1}{2}$

Now calculate $S_6$:

$S_6 = \frac{6}{2} \left( 2\left(-\frac{5}{4}\right) + (6-1)\frac{1}{2} \right)$ $S_6 = 3 \left( 2 \times -\frac{5}{4} + 5 \times \frac{1}{2} \right)$ $S_6 = 3 \left( -\frac{10}{4} + \frac{5}{2} \right)$ $S_6 = 3 \left( -\frac{10}{4} + \frac{10}{4} \right)$ $S_6 = 3 \times 0 = 0$

Thus, the sum of the first 6 terms $S_6 = 0$.

Would you like more details on this, or do you have any questions?

Related Questions:

1. How is the common difference in an arithmetic progression calculated?
2. What is the general formula for the sum of the first $n$ terms of an arithmetic progression?
3. Can the sum of terms in an arithmetic progression be zero for other values of $n$?
4. How does changing the common difference affect the sum of an arithmetic progression?
5. What is the sum of the first 10 terms of this arithmetic progression?

Tip: In an arithmetic progression, when the sum of certain terms equals zero, this often indicates symmetry in the sequence values around the midpoint.