Given below is an arithmetic progression. What is Sg for this arithmetic progression?
-5/4,-3/4,-1/4,..
S6=

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.

Given below is an arithmetic progression. What is Sg for this arithmetic progression? -5/4,-3/4,-1/4,.. S6=

Calculate the sum of the first 6 terms (S6) of the arithmetic progression -5/4, -3/4, -1/4,... using the arithmetic series summation formula. This step-by-step solution demonstrates how to find the common difference and apply the sum formula. Suitable for students in Grades 9-11.

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