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## Geometric Sequences and Series

Given our generic geometric sequence…

…we can look at it as a series.

As we can see, the only difference between a sequence and a series is that a sequence is a list of numbers and a series is a sum of numbers.

There exists a formula that can add a finite list of numbers and a formula for an infinite list of numbers. Here are the formulas…

…where Sn is the sum of the first n numbers, a1 is the first number in the sequence, r is the common ratio of the sequence, and -1

Example 1: Find the sum of the first 7 terms of the sequence below.

n12345 . . . Term124816 . . .

The sum formula requires us to know the first term [a1], the common ratio [r], and the number of terms [n]. We know the first term is 1. The common ratio is 2. The number of terms is 7. Plugging this information into the formula give us this.

So, the sum of the first 7 terms is 127.

Example 2: Add the first 10 terms of the sequence below.

n12345 . . . Term0.010.060.362.1612.96 . . .

We can see a1 = 0.01, r = 6 and we were told n = 10. We would then plug those numbers into the formula and get this.

So, the sum of the first 10 terms is 120,932.35.

ideo: Sum of a Finite Geometric Series uizmaster: Finding the Sum of a Finite Series

Example 3: Add the infinite series 16 + (-8) + 4 + (-2) + 1 + …

The only way we can add an infinite series is for two conditions to be met: a) it has to be a geometric series and b) the common ratio has to be greater than -1 but less than 1.

Looking at the series, we can see that there is a common ratio. This means it is geometric. Since the common ratio is -1/2 and it falls between -1 and 1, we can use the sum formula. We will use a1 = 16 and r = -1/2.

This means the entire infinite series is equal to 102/3.

Example 4: Add the infinite sum 27 + 18 + 12 + 8 + …

We need to check the conditions to see if we can use the infinite sum formula. It does have a common ratio. It is 2/3. Since 2/3 is less than 1 and greater than -1, we can use the formula, like this.

ideo: Sum of an Infinite Geometric Series uizmaster: Finding the Sum of an Infinite Series

Arithmetic Sequence Formula – ChiliMath – The Arithmetic Sequence Formula. If you wish to find any term (also known as the {{nth}} term) in the arithmetic sequence, the arithmetic sequence formula should help you to do so. The critical step is to be able to identify or extract known values from the problem that will eventually be substituted into the formula itself.Find a8 when a1 = -10, d = -3. Answer-31 11-34 14 3 points Question 16 Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. Find a21 when a1 = 28, d = -5. Answer-77 128-100-72 3 points Question 17 Write a formula for theFor example, the sequence 3, 6, 9, 12, 15, 18, 21, 24… is an arithmetic progression having a common difference of 3. The formulas applied by this arithmetic sequence calculator can be written as explained below while the following conventions are made: – the initial term of the arithmetic progression is marked with a 1;

Question 1 Write the first four terms of the sequence – Free Geometric Sequences calculator – Find indices, sums and common ratio of a geometric sequence step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy.Search the world's information, including webpages, images, videos and more. Google has many special features to help you find exactly what you're looking for.👉 Learn how to find the nth term of a geometric sequence. A sequence is a list of numbers/values exhibiting a defined pattern. A number/value in a sequence

Arithmetic Sequence Calculator – a n = a 1 + (n-1)d, where a 1 is the first term and d is the common difference. The following diagrams give an arithmetic sequence and the formula to find the n th term. Scroll down the page for more examples and solutions. Arithmetic Sequences This video covers identifying arithmetic sequences and finding the nth term of a sequence. Itbut we don't know what the 15th term is so we use the first equation to solve for the 15th term. let a1=5 , d=4 and the number you need is the 15th term so you have 61. so the 15th term is 61. so we use the sum equation. Sn=15/2 * (5+61) = 495. so use the same concepts for #3. 12th term is -46 #4Identify the Sequence 4 , 8 , 16 , 32, , , This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by gives the next term. In other words, . Geometric Sequence: This is the form of a geometric sequence. Substitute in the values of and .