Sum Of Arithmetic Sequence: 131 + 147 + 143 + ... + -5

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Hey guys! Let's dive into how to calculate the sum of an arithmetic sequence. We've got a fun one here: 131 + 147 + 143 + 139 + ... + -5. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you'll be a pro in no time.

Understanding Arithmetic Sequences

First, let's make sure we're all on the same page. An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. Identifying this difference is key to solving our problem.

In our sequence, 131, 147, 143, 139, ..., -5, we can see that the terms are changing, but not in a straightforward way. It seems like there might be two interwoven sequences here – one increasing and one decreasing. Let's separate them to make it clearer:

  • Sequence 1: 131, 143, ...
  • Sequence 2: 147, 139, ...

Now, let's find the common difference for each sequence:

  • For Sequence 1 (131, 143, ...), the common difference is 143 - 131 = 12.
  • For Sequence 2 (147, 139, ...), the common difference is 139 - 147 = -8.

Okay, now we've got two arithmetic sequences with their common differences. This is a crucial step because we need this information to calculate the sum. Remember, the main keyword here is arithmetic sequence, so understanding its properties is super important!

Breaking Down the Problem

To find the sum of the entire sequence, we can find the sum of each individual sequence (Sequence 1 and Sequence 2) and then add those sums together. This makes the problem much more manageable. So, let's focus on each sequence separately.

The formula for the sum of an arithmetic series is:

S = (n/2) * [2a + (n - 1)d]

Where:

  • S is the sum of the series.
  • n is the number of terms.
  • a is the first term.
  • d is the common difference.

Before we can use this formula, we need to find 'n' (the number of terms) for each sequence. We also need to figure out how many terms are in each sequence before we reach -5. This requires a bit more digging!

Finding the Number of Terms (n)

Let's find the number of terms in Sequence 1 (131, 143, ...) that eventually contributes to the combined sequence leading to -5. To do this, we need to consider how the two sequences interleave. We will use the arithmetic sequence formula to find the nth term:

Tn = a + (n - 1)d

Where:

  • Tn is the nth term.
  • a is the first term.
  • n is the term number.
  • d is the common difference.

We need to figure out which terms from each sequence combine to get us close to -5. Let's analyze further terms in both sequences.

Sequence 1: 131, 143, 155, 167,... (Common difference = 12)

Sequence 2: 147, 139, 131, 123, 115, 107, 99, 91, 83, 75, 67, 59, 51, 43, 35, 27, 19, 11, 3, -5,... (Common difference = -8)

Notice that Sequence 2 is decreasing and will eventually include -5. Sequence 1, however, is increasing. We need to find how Sequence 1 contributes to the overall sequence sum before Sequence 2 reaches -5.

To find the term number in Sequence 2 that is -5, we use the formula:

-5 = 147 + (n - 1)(-8)

Solving for n:

-5 = 147 - 8n + 8

-160 = -8n

n = 20

So, -5 is the 20th term in Sequence 2. Now we need to figure out how many terms from Sequence 1 are included up to this point. This is a tricky part, guys, so pay close attention!

Let's list out the terms of both sequences until we reach the 20th term of Sequence 2. Since the sequences interleave, we'll see how many terms of Sequence 1 are included:

Combined Sequence: 131, 147, 143, 139, 155, 131, 167, 123, 179, 115, 191, 107, 203, 99, 215, 91, 227, 83, 239, 75, ...

By manually listing out enough terms of both sequences and interleaving them, or by carefully considering the pattern, we can determine that to reach the 20th term of the second sequence (which is -5), we will have considered 10 terms from the first sequence.

This is a critical step – we found that Sequence 1 has 10 terms and Sequence 2 has 20 terms within the overall sequence we're summing.

Calculating the Sum of Each Sequence

Now that we know the number of terms for each sequence, we can use the formula for the sum of an arithmetic series:

S = (n/2) * [2a + (n - 1)d]

For Sequence 1 (131, 143, ...):

  • n = 10
  • a = 131
  • d = 12

S1 = (10/2) * [2(131) + (10 - 1)(12)]

S1 = 5 * [262 + 108]

S1 = 5 * 370

S1 = 1850

For Sequence 2 (147, 139, ...):

  • n = 20
  • a = 147
  • d = -8

S2 = (20/2) * [2(147) + (20 - 1)(-8)]

S2 = 10 * [294 - 152]

S2 = 10 * 142

S2 = 1420

Finding the Total Sum

Finally, we can find the total sum by adding the sums of the two sequences:

Total Sum = S1 + S2

Total Sum = 1850 + 1420

Total Sum = 3270

Conclusion

So, the sum of the arithmetic sequence 131 + 147 + 143 + 139 + ... + -5 is 3270! That was a bit of a journey, but we got there by breaking down the problem into smaller, manageable parts. Remember the main keyword, arithmetic sequence, and how we used its properties to solve this. Keep practicing, guys, and you'll master these sequences in no time!