source : mathguide.com
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
How to Identify a Term in a Geometric Sequence When You – If your pre-calculus teacher gives you any two nonconsecutive terms of a geometric sequence, you can find the general formula of the sequence as well as any specified term. For example, if the 5th term of a geometric sequence is 64 and the 10th term is 2, you can find the 15th term. Just follow […]If the initial term of an arithmetic sequence is a 1 and the common difference of successive members is d, then the nth term of the sequence is given by: a n = a 1 + (n – 1)d The sum of the first n terms S n of an arithmetic sequence is calculated by the following formula:For 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;
Arithmetic Sequence Calculator – MiniWebtool – Using the same geometric sequence above, find the sum of the geometric sequence through the 3 rd term. EX: 1 + 2 + 4 = 7. 1 × (1-2 3) 1 – 2 = -7-1 = 7: Fibonacci Sequence. A Fibonacci sequence is a sequence in which every number following the first two is the sum of the two preceding numbers. The first two numbers in a Fibonacci sequence area 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. ItFirst, we would identify the common difference. We can see the common difference is 4 no matter which adjacent numbers we choose from the sequence. To find the next number after 19 we have to add 4. 19 + 4 = 23. So, 23 is the 6th number in the sequence. 23 + 4 = 27; so, 27 is the 7th number in the sequence, and so on…
Arithmetic Sequence Calculator – Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Arithmetic Sequence Calculator Find indices, sums and common diffrence of an arithmetic sequence step-by-step Common Difference Next Term N-th Term Value given Index Index givenAn arithmetic sequence is any list of numbers that differ, from one to the next, by a constant amount. For example, the list of even numbers, ,,,, … is an arithmetic sequence, because the difference from one number in the list to the next is always 2. If you know you are working with an arithmetic sequence, you may be asked to find the very next term from a given list. You may also be askedGiven an arithmetic sequence with the first term a 1 and the common difference d , the n th (or general) term is given by a n = a 1 + ( n − 1 ) d . Example 1: Find the 27 th term of the arithmetic sequence 5 , 8 , 11 , 54 , .