Find the sum of this sequence 101,  95, 89, ....  -67.
*
2 puntos
393
483
493
593

To find the sum of the arithmetic sequence \(101, 95, 89, \ldots, -67\), we can again use the sum formula for an arithmetic sequence:

\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]

### Step 1: Identify the given values
- First term \( a_1 = 101 \)
- Last term \( a_n = -67 \)
- Common difference \( d = 95 - 101 = -6 \)

### Step 2: Find the number of terms (\( n \))
The \( n \)-th term of an arithmetic sequence is given by:

\[
a_n = a_1 + (n - 1) \times d
\]

Substitute the known values:

\[
-67 = 101 + (n - 1) \times (-6)
\]

Solve for \( n \):

\[
-67 - 101 = (n - 1) \times (-6)
\]
\[
-168 = (n - 1) \times (-6)
\]
\[
n - 1 = \frac{-168}{-6} = 28
\]
\[
n = 29
\]

So, there are 29 terms in the sequence.

### Step 3: Calculate the sum of the sequence
Now, use the sum formula:

\[
S_{29} = \frac{29}{2} \times (101 + (-67)) = \frac{29}{2} \times 34 = 29 \times 17 = 493
\]

So, the sum of the sequence is **493**.

Would you like more details or have any questions?

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Here are five related questions:
1. What would be the sum of the first 15 terms of this sequence?
2. How do you find the common difference when only given a few terms of a sequence?
3. What is the sum of an arithmetic sequence where the first term is 50, the common difference is -4, and the last term is 2?
4. How does the sign of the common difference affect the sum of the sequence?
5. Can the sum formula be used if the sequence is infinite? Why or why not?

**Tip:** Remember that a negative common difference means the terms of the sequence are decreasing.

Learn how to find the sum of the arithmetic sequence 101, 95, 89, ..., -67 using the sum formula for arithmetic sequences. This problem covers key concepts including arithmetic sequences, common differences, and sequence summation. Suitable for high school students.

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