Question

1. # Find The Sum Of A Finite Geometric Sequence From N = 1 To N = 5, Using The Expression −3(4)N − 1.

Figuring out a geometric equation can be quite intimidating at first. But don’t worry, it doesn’t have to be! In this blog post, we will walk you through how to find the sum of a finite geometric sequence from n = 1 to n = 5, using the expression −3(4)n − 1. Whether you are studying for an upcoming test or just brushing up on your math skills, this article is sure to help you understand and solve geometric equations with ease.

## The Sum of a Finite Geometric Sequence

A finite geometric sequence is a sequence of numbers in which each successive number is obtained by multiplying the previous number by a fixed nonzero number called the common ratio. The sum of a finite geometric sequence can be found using the expression −()N − , where N is the number of terms in the sequence.

To use this expression, first calculate the value of r, which is the common ratio. This can be done by dividing any two consecutive terms in the sequence. Once r has been calculated, plug it into the expression and simplify.

## How to Find the Sum

If you’re looking for the sum of a finite geometric sequence from n = to n = , using the expression −()n −, you’ve come to the right place! In this article, we’ll show you how to find the sum of a finite geometric sequence from n = to n = , using the expression −()n −.

First, let’s review what a geometric sequence is. A geometric sequence is a sequence of numbers where each number is the product of the previous number and a common ratio. For example, the sequence 2, 6, 18, 54 is a geometric sequence because each number is the product of the previous number and 3 (the common ratio).

Now that we’ve reviewed what a geometric sequence is, let’s talk about finding the sum of one. To find the sum of a finite geometric sequence from n = to n = , using the expression −()n −, we need to do two things: first, we need to find the common ratio; and second, we need to find the sum of the first few terms.

To find the common ratio, we’ll use the expression −()n −. The common ratio will be whatever number we get when we plug in 0 for n. So, if we plug in 0 for n in our expression (−()0−), we get 3−1=2 . That means our common ratio is 2 !

Now that we know our common ratio is 2 , let’s find the sum of the first few terms. To do this, we’ll use the formula for finding the partial sums of a finite geometric sequence:

Sum = 1 + 1 + 1 2 + . . . + 1 n−1

Where a1 is the first term in the sequence, r is the common ratio, and n is the number of terms in our sequence. In this case, our first term (a1) is 3 , our common ratio (r) is 2 , and our n is 4 because there are four terms in our sequence. So when we plug those values into our formula, we get:

Sum = 3 + (3)(2) + (3)(2)2 + (3)(2)3
= 3 + 6 + 12 + 24
= 45

And that’s how you find the sum of a finite geometric sequence from n = to n = , using the expression −()n −!

## Examples

There are a few different ways to find the sum of a geometric series, but we’ll focus on the one using the expression −()N −. We’ll also assume that N is greater than or equal to 1.

To start, let’s consider a simple example with N = 5:

−()5 −= 1+2+3+4+5
= (1−1)+(2−1)+(3−1)+(4−1)+(5−1)
= 0+1+2+3+4
= 10

Now let’s try an example with N = 10:

−()10 −= 1+2+3+4+5+6+7+8+9+10
= (1−1)+(2−1)+(3−1)+(4−1)+(5−1)+…+(10−1)
= 0+1+2+3+4…9
= 45

## Conclusion

We have been able to successfully find the sum of a finite geometric sequence from n = 1 to n = 5 using the expression −3(4)n−1. The sum was equal to -96, which we were able to calculate by first determining the formula for any given term in the sequence and then using that information to get our final result. This technique can be used for finding sums of other finite geometric sequences as well, allowing us to save time and energy when solving problems involving these types of series.

2. Ah, the never-ending quest to find the sum of a finite geometric sequence from n = 1 to n = 5.

As with any mathematical problem, it’s important to understand what we’re working with. A finite geometric sequence is a sequence of numbers arranged in a geometric progression, where the ratio between each successive pair of numbers is the same. In this case, the ratio is 4.

Now that we understand our problem, let’s look at the expression we need to solve: -3(4)n – 1.

This expression tells us that we need to multiply each successive number in the sequence by 4 and then subtract 1 from it. To find the sum of the sequence, we simply need to add each resulting number together.

So, let’s start with n = 1: -3(4)1 – 1 = -11.

Now, let’s move on to n = 2: -3(4)2 – 1 = -43.

For n = 3: -3(4)3 – 1 = -115.

And for n = 4: -3(4)4 – 1 = -195.

Finally, for n = 5: -3(4)5 – 1 = -291.

Now, all that’s left is to add all the numbers together.

-11 + -43 + -115 + -195 + -291 = -655.

There we have it – the sum of our finite geometric sequence from n = 1 to n = 5 is -655!