What is the sum of the geometric sequence 1/4 16 If there are 8 terms?

Geometric Sequences: Let’s Add ‘Em Up! (Especially That 1/4, 1, 4, 16 Thing…)

Okay, so geometric sequences. They might sound intimidating, but they’re really just a list of numbers with a cool pattern. You see them pop up all over the place, from calculating interest on your savings to modeling how a population grows. The trick is understanding how they work. Today, we’re going to tackle a specific one and figure out how to add up its first eight terms: 1/4, 1, 4, 16… and so on.

So, What’s the Deal with Geometric Sequences?

Basically, a geometric sequence is a series where you get the next number by multiplying the previous one by the same amount every time. Think of it like this: you start with a number, and then it just keeps growing (or shrinking!) at a consistent rate. That consistent rate is what we call the “common ratio.”

Finding That “Common Ratio” – It’s Easier Than You Think

In our example sequence (1/4, 1, 4, 16…), we need to figure out what that common ratio is. What are we multiplying by each time? The easiest way to find out is to just divide any term by the term before it. So:

• 1 divided by 1/4? That’s 4.
• 4 divided by 1? Yep, still 4.
• 16 divided by 4? You guessed it: 4.

So, our common ratio (usually shown as ‘r’) is a solid 4. We’re multiplying by 4 each time to get the next number in the sequence.

The Magic Formula: Adding Up a Bunch of Numbers the Fast Way

Now, here’s where the real fun begins. Instead of manually adding 1/4 + 1 + 4 + 16 + … (eight times!), we can use a nifty formula. It’s like a shortcut for adding up geometric sequences:

Sn = a1 * (1 – rn) / (1 – r)

Let’s break that down:

• Sn is what we’re trying to find: the sum of the first ‘n’ terms.
• a1 is the first term in the sequence (in our case, that’s 1/4).
• r is our common ratio (which we already know is 4).
• n is the number of terms we want to add up (we’re going for 8 terms).

Let’s Plug in the Numbers and See What Happens

Alright, time to put our formula to work! We know:

• a1 = 1/4
• r = 4
• n = 8

So, let’s substitute those values:

S8 = (1/4) * (1 – 48) / (1 – 4)

Now, let’s simplify:

S8 = (1/4) * (1 – 65536) / (-3)


S8 = (1/4) * (-65535) / (-3)


S8 = (1/4) * 21845


S8 = 5461.25

The Grand Finale: The Sum is…

So, after all that calculating, we find that the sum of the first 8 terms of the geometric sequence 1/4, 1, 4, 16… is 5461.25. Not bad, huh?

The Big Picture

• Geometric sequences have a constant multiplier between terms – the common ratio.
• There’s a formula that makes adding up a bunch of terms super easy.
• Take your time, plug in the numbers carefully, and you’ll get the right answer.

Once you get the hang of geometric sequences and their formulas, you’ll start seeing them everywhere. And you’ll be able to impress your friends with your math skills. Go forth and calculate!