Finding The Sum Of The First 6 Terms Of Geometric Sequences

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Hey guys! Today, we're diving into the fascinating world of geometric sequences. We're going to break down how to find the sum of the first six terms of a geometric sequence. Don't worry, it's not as intimidating as it sounds! We'll take it step by step and you'll be a pro in no time. So, let's get started and make math fun!

Understanding Geometric Sequences

Before we jump into the calculations, let's quickly recap what geometric sequences are all about. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant value. This constant value is called the common ratio, often denoted as 'r'.

To put it simply, think of it like this: you start with a number, and to get the next number in the sequence, you multiply by the same value every time. This consistent multiplication is what defines a geometric sequence.

For example, if we have a sequence that starts with 2 and has a common ratio of 3, the sequence would look like this: 2, 6, 18, 54, and so on. Each term is three times the previous term. See how it works?

Knowing this basic concept is crucial because it lays the foundation for everything else we're going to do. Understanding the common ratio is especially important because it's a key ingredient in finding the sum of a geometric sequence. We'll be using it a lot, so make sure you've got a good handle on what it means!

Formula for the Sum of a Geometric Sequence

Now, here’s the magic formula we’ll use to find the sum of the first 'n' terms of a geometric sequence:

Sn = a(1 - r^n) / (1 - r)

Where:

  • Sn is the sum of the first 'n' terms.
  • 'a' is the first term of the sequence.
  • 'r' is the common ratio.
  • 'n' is the number of terms we want to sum up.

This formula might look a bit scary at first, but trust me, it's not that bad once you break it down. Let’s go through each part:

  • Sn: This is what we’re trying to find – the sum of the first 'n' terms. It’s the answer we're looking for!
  • a: This is the easiest one. It's simply the first number in the sequence. You just need to identify the starting point.
  • r: This is the common ratio we talked about earlier. It’s the number you multiply by to get from one term to the next. Finding 'r' is usually one of the first steps in solving these problems.
  • n: This is the number of terms you want to add up. In our case, we're looking for the sum of the first 6 terms, so 'n' will be 6.

So, with this formula in our toolkit, we can tackle any problem that asks us to find the sum of a geometric sequence. Remember, the key is to identify 'a', 'r', and 'n', and then plug them into the formula. We'll do plenty of examples to make sure you've got it down!

Let's Solve Some Problems!

Alright, let's put our knowledge to the test and solve some problems. We'll tackle each sequence one by one, making sure we understand each step. Remember, the goal is to find the sum of the first 6 terms for each sequence. So, let's roll up our sleeves and get started!

a) 1st term = 4 and 2nd term = 8

Okay, for our first sequence, we're given that the first term (a) is 4 and the second term is 8. The first thing we need to do is find the common ratio (r). Remember, the common ratio is what we multiply by to get from one term to the next.

To find 'r', we can divide the second term by the first term:

r = 8 / 4 = 2

So, our common ratio is 2. Now we have:

  • a = 4
  • r = 2
  • n = 6 (since we want the sum of the first 6 terms)

Now we just plug these values into our formula:

S6 = 4 * (1 - 2^6) / (1 - 2)

Let's break this down step by step:

  1. 2^6 = 64
  2. 1 - 64 = -63
  3. 1 - 2 = -1
  4. S6 = 4 * (-63) / (-1)
  5. S6 = -252 / -1
  6. S6 = 252

So, the sum of the first 6 terms of this sequence is 252. Awesome! We've solved our first one. See, it's not so bad once you break it down. We identified the key values, plugged them into the formula, and followed the steps. Let's keep going!

b) 1st term = 5 and 2nd term = 15

Next up, we have a sequence where the first term (a) is 5 and the second term is 15. Just like before, our first mission is to find the common ratio (r). To do that, we'll divide the second term by the first term:

r = 15 / 5 = 3

So, our common ratio is 3. Now we know:

  • a = 5
  • r = 3
  • n = 6 (we're still looking for the sum of the first 6 terms)

Time to plug these values into our trusty formula:

S6 = 5 * (1 - 3^6) / (1 - 3)

Let's tackle this one step by step as well:

  1. 3^6 = 729
  2. 1 - 729 = -728
  3. 1 - 3 = -2
  4. S6 = 5 * (-728) / (-2)
  5. S6 = -3640 / -2
  6. S6 = 1820

Therefore, the sum of the first 6 terms of this sequence is 1820. Another one down! We’re on a roll. You’re probably starting to see the pattern now, which is fantastic. Let’s keep the momentum going!

c) 1st term = 64 and 2nd term = 32

Alright, moving on to our third sequence, the first term (a) is 64 and the second term is 32. You know the drill – let's find that common ratio (r) first. We divide the second term by the first term:

r = 32 / 64 = 1/2 = 0.5

So, our common ratio is 0.5 (or 1/2). This tells us that the sequence is getting smaller with each term, which is interesting. Now we have:

  • a = 64
  • r = 0.5
  • n = 6 (still finding the sum of the first 6 terms)

Let's plug these into the formula and see what we get:

S6 = 64 * (1 - 0.5^6) / (1 - 0.5)

Let's break it down:

  1. 0.5^6 = 0.015625
  2. 1 - 0.015625 = 0.984375
  3. 1 - 0.5 = 0.5
  4. S6 = 64 * (0.984375) / 0.5
  5. S6 = 62.99999999999999 / 0.5
  6. S6 = 126 (approximately)

So, the sum of the first 6 terms of this sequence is approximately 126. Notice that we got a decimal in our calculations, but the final answer is a whole number (after rounding). This is a great example of how geometric sequences can behave differently depending on the common ratio. We're doing great! Only one more to go.

d) 1st term = 243 and 2nd term = 81

Last but not least, we have a sequence where the first term (a) is 243 and the second term is 81. Let’s find that common ratio (r) by dividing the second term by the first term:

r = 81 / 243 = 1/3 ≈ 0.333

So, the common ratio is 1/3 (or approximately 0.333). This is another sequence that's getting smaller with each term. We now know:

  • a = 243
  • r = 1/3
  • n = 6

Let's plug these values into the formula one last time:

S6 = 243 * (1 - (1/3)^6) / (1 - 1/3)

This one looks a bit trickier with the fraction, but we can handle it. Let's break it down:

  1. (1/3)^6 = 1 / 729
  2. 1 - (1/729) = 728 / 729
  3. 1 - 1/3 = 2/3
  4. S6 = 243 * (728/729) / (2/3)
  5. S6 = (243 * 728 * 3) / (729 * 2)
  6. S6 = 212582 / 1458
  7. S6 = 364

So, the sum of the first 6 terms of this sequence is 364. We did it! We've successfully found the sum of the first 6 terms for all the given geometric sequences. Give yourselves a pat on the back – you've earned it!

Conclusion

Wow, guys, we've covered a lot today! We've revisited the concept of geometric sequences, learned about the common ratio, and, most importantly, mastered the formula for finding the sum of the first 'n' terms. We tackled four different sequences and saw how the formula works in action.

Remember, the key to success with geometric sequences is to break down the problem into smaller steps. First, identify the first term (a) and the common ratio (r). Then, decide how many terms you need to sum up (n). Finally, plug those values into the formula and do the math carefully. With a little practice, you'll become super confident in solving these types of problems.

Geometric sequences pop up in all sorts of places, from finance to physics, so understanding them is a valuable skill. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!