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Formula for the sum of geometric progressions
A geometric sequence is a sequence that keeps multiplying the number in the first term.
【Example】
$$3,\ 3\times4,\ 3\times4^2,\ 3\times4^3,\cdots,\ 3\times4^{n}$$
Expressed numerically, it looks like the above.
The above sequence is a geometric sequence with the first term \(3\) and the common ratio \(4\).
And the formula for the sum of geometric progressions is given below.
The sum of the geometric progression of the first term \(a\), the common ratio \(r\), and the number of terms \(n\) is
\begin{eqnarray}
S_n=\begin{cases}
\displaystyle \frac{a(1-r^n)}{1-r}=\displaystyle \frac{a(r^n-1)}{r-1}\ \ &(r\neq 1)&\ \
na&(r=1)&
\end{cases}\end{eqnarray}
The sum of the geometric progression is used properly by the common ratio \(r\).
\(r≠1\) or \(r=1\).
Formula memorization and puns
The official pun is "Ii visits, English also visits the n-th power"is.
$$S_n=\displaystyle \frac{a(1-r^n)\{Eng also comes to the n-th power\}}{1-r\{Ii comes to the \}}$$
I introduced the puns, but there is also a way to memorize the formula, which will be explained next, to prove the sum of geometric progressions.
This is because once you have learned how to prove it, you can easily guide it.
Use whoever you like!
Prove the Sum of Geometric Sequences
Now let's prove the sum of geometric progressions.
The sum of the geometric progression must be used properly according to the common ratio \(r\).
\begin{eqnarray}
S_n=\begin{cases}
\displaystyle \frac{a(1-r^n)}{1-r}=\displaystyle \frac{a(r^n-1)}{r-1}\ \ &(r\neq 1)&\ \
na&(r=1)&
\end{cases}\end{eqnarray}
when r=1
Let \(S_n\) be the sum from the first term to the \(n\)th term of the geometric progression of the first term \(a\) and the common ratio \(1\), then it can be written as follows .
$$S_n=a+a+a\cdots a+a=na$$
when r≠1
Let \(S_n\) be the sum of the geometric progression of the first term \(a\), the common ratio \(r\), and the number of terms \(n\).
$$S_n=a+ar+ar^2+ar^3\cdots ar^{n-2}+ar^{n-1}\cdots①$$
Here, both sides of the formula ① are multiplied by \(r\).
$$rS_n=ar+ar^2+ar^3+ar^4\cdots ar^{n-2}+ar^{n-1}+ar^n\cdots②$$
It can be calculated as follows from \(①-②\)
\begin{eqnarray}
S_n&=&a+ar+ar^2+ar^3\cdots ar^{n-2}+ar^{n-1}\\
-rS_n&=&\ \ \ \ -ar-ar^2-ar^3\cdots ar^{n-2}-ar^{n-1}-ar^n)\\
(1-r)S_n&=&a-ar^{n}\\
S_n&=&\displaystyle \frac{a(1-r^n)}{1-r}
\end{eqnarray}
From the above, the sum of the geometric progression of the first term \(a\), the common ratio \(r=1\), and the number of terms \(n\) is
\begin{eqnarray}
S_n=\begin{cases}
\displaystyle \frac{a(1-r^n)}{1-r}=\displaystyle \frac{a(r^n-1)}{r-1}\ \ &(r\neq 1)&\ \
na&(r=1)&
\end{cases}\end{eqnarray}
I want to read it together
![[Number B] Sum of geometric progressions | Formulas, how to remember, proofs, etc. explained at once | Tomlab (1)](https://i0.wp.com/rikeinvest.com/wp-content/uploads/2022/07/等差数列の和の公式-500x263.webp "[Number B] Sum of geometric progressions | Formulas, how to remember, proofs, etc. explained at once | Tomlab (1)")
Proof of the formula for the sum of geometric progressionsWe prove the following formula for the sum of geometric progressions.\begin{eqnarray} S_n=\begin{cases}\displaystyle \frac {a(1-r^...
The reason why the sum of up to n-1 terms is the n-th power
In the question, ask, "It is the sum of a geometric progression, but why does \(r^n\) appear in the formula even though it is the sum of the first term to the \(n-1\) term?" will be
You can find the answer by looking at the proof.
Since the difference between \(S_n\) and \(rS_n\) is calculated, \(r^n\) appears even though the calculation is up to the \(n-1\) term.
sum of geometric progressions using sigma (Σ)
Now let's think about sums of geometric progressions using sigma.
\(\displaystyle \sum_{k=1}^n ar^{k-1}\)
The meaning of this expression is "the sum when \(1 to n\) is substituted for \(k\)".
\begin{eqnarray}
k=1\rightarrow ar^0=a\\
k=2\rightarrow ar^{2-1}=ar\\
k=3\rightarrow ar^{3-1}=ar^2\\
\cdots\\
k=n-1 \rightarrow ar^{(n-1)-1}=ar^{n-2}\\
k=n\rightarrowar^{n-1}\\
\end{eqnarray}
You can calculate it like this.
I will add them all!That's what the sigma formula means.
\(\displaystyle \sum_{k=1}^n ar^{k-1}\)
In other words, it is the sum of geometric progressions.
How to find the sum of a geometric progression using sigma
So how do you find the sum of geometric progressions from the sigma formula?I have a question.
The answer is to find the first term \(a\), the common ratio \(r\), and the number of terms \(n\) from the sigma equation.
Substitute into the formula for the sum of geometric progressions!is the answer.
For example, what about the formula below:
$$\displaystyle \sum_{k=1}^8 5\times 2^{k+1}$$
Let's find it in order of the first term, the common ratio, and the number of terms.
first term
The first term is the first term, so it is the value when \(k=1\).
So \(5\times 2{1+1}=5\times2^2=20\) and \(a=20\).
common ratio
Next, find the common ratio.
\begin{eqnarray}
k=1\rightarrow 5\times2^2=20\\
k=2\rightarrow 5\times2^3=40\\
k=3\rightarrow 5\times2^4=80\\
\cdots
\end{eqnarray}
You can see that it is \(\times2\) as above.
So the common ratio is \(r=2\).
number of terms
Finally, consider the number of terms.
The problem is \(\displaystyle \sum_{k=1}^8 5\times 2^{k+1}\).
The meaning of the sigma (\displaystyle \sum_{k=1}^8 ) part is from \(k=1\) to \(k=2\), \(k=3\) and increasing by 1 Increase to \(8\).And I'll take it allis what it means.
So the number of terms is from \(1\) to \(8\), so \(n=8\).
To summarize the above,
First term\(a=20\), common ratio\(r=2\), number of terms\(n=8\).
Since the common ratio is \(r≠1\), substitute it into the following formula.
\(S_n=\displaystyle \frac{a(1-r^n)}{1-r}\)
Substituting and calculating,
\begin{eqnarray}
S_n&=&\displaystyle \frac{20(1-2^8)}{1-2}\\
&=&\displaystyle \frac{20(1-256)}{-1}\\
&=&20\times255\\
&=&5100
\end{eqnarray}
From the above,
\(\displaystyle \sum_{k=1}^8 5\times 2^{k+1}=5100\)
when k=0
One thing to note about sigma (\(\displaystyle \sum_{k=1}^n\)).
The point is that it may start from \(k=0\).
For example, let's say the previous problem started with \(k=0\) as shown below.
$$\displaystyle \sum_{k=0}^8 5\times 2^{k+1}$$
There are two ways to solve it.
- Find the first term, the common ratio, and the number of terms and substitute
- Add the value of \(k=0\) and the value of \(\displaystyle \sum_{k=1}^8 5\times 2^{k+1}\)
Let's go through them one by one.
Find the first term, common ratio, and number of terms
First, let's look at how to find the first term, the common ratio, and the number of terms, which are the most common solutions.
The first term is \(k=0\), so \(5\times 2^1=10\)
The common ratio is \(r=2\)
The number of terms is \(0 to 8\), so \(n=9\)
Substituting into the formula for the sum of geometric progressions, we get
\begin{eqnarray}
S_n&=&\displaystyle \frac{10(1-2^9)}{1-2}\\
&=&\displaystyle \frac{10(1-512)}{-1}\\
&=&10\times511\\
&=&5110
\end{eqnarray}
It becomes.
add the value of \(k=0\)
Next, let's see how to add \(k=0\).
If you know the value of \(\displaystyle \sum_{k=1}^8 5\times 2^{k+1}\), this method is easy.
From the property of sigma, the following transformation of the formula holds.
$$\displaystyle \sum_{k=0}^8 5\times 2^{k+1}=5\times 2^{0+1}+\displaystyle \sum_{k=1}^8 5\times 2 ^{k+1}$$
So adding \(0\times 8=5\) to \(\displaystyle \sum_{k=2}^1 5\times 2^{k+10}\) gives the answer.
\(\displaystyle \sum_{k=1}^8 5\times 2^{k+1}=5100\) is found because it was explained earlier.
If you add \(10\) to this, you can easily get \(5110\).
Situations where this formula can be used
Examples of situations where this formula can be used are:
[Problem] Find the next value
(1) \(\displaystyle \sum_{k=1}^8 5\times 2^{k+1}\)
(2) \(\displaystyle \sum_{k=0}^8 5\times 2^{k+1}\)
In this case, (2) does not need to be calculated.
It can be obtained by adding the value (\(1\times 0^{5+2}=0\)) when \(k=1\) to the value obtained in (10).
sum of infinite geometric progressions
I will explain the sum of infinite geometric progressions.
Written in sigma, the formula looks like this:
$$\displaystyle \sum_{k=1}^{\infty}ar^{k-1}=a+ar+ar^2+\cdots$$
To find this sum, we can use the formula for the sum of geometric progressions.
However, even if I know the first term \(a\) and the common ratio \(r\), the number of terms \(n=\infty\) makes it impossible to calculate.
So we use \(\displaystyle\lim_{n\to\infty}\) to express it.
\begin{eqnarray}
&&\displaystyle \sum_{k=1}^{\infty}ar^{k-1}\\
&=&\displaystyle\lim_{n\to\infty}S_n\\
&=&\displaystyle\lim_{n\to\infty}\displaystyle \frac{a(1-r^n)}{1-r}
\end{eqnarray}
Here \(\displaystyle\lim_{n\to\infty}r^n\) is the point.
In conclusion \(-1
\(\displaystyle\lim_{n\to\infty}r^n = 0\).
For example, if \(r=0.5\),
\begin{eqnarray}
n=1&\rightarrow&r^1=0.5\\
n=2&\rightarrow&r^2=0.25\\
n=3&\rightarrow&r^3=0.125\\
\cdots
\end{eqnarray}
and finally \(\displaystyle\lim_{n\to\infty}r^n = 0\).
If the absolute value of \(r\) is less than \(1\) (\(-1
\(\displaystyle\lim_{n\to\infty}r^n\) diverges to infinity when the absolute value of \(r\) is greater than or equal to \(1\).
From the above, the sum of infinite geometric progressions can be summarized as follows.
\begin{eqnarray}
\displaystyle \sum_{k=1}^{\infty} ar^{k-1}=\displaystyle\lim_{n\to\infty}\displaystyle \frac{a(1-r^n)}{1- r}=
\begin{cases}\displaystyle \frac{a(1-0)}{1-r}= \displaystyle \frac{a}{1-r}\ (|r|<1)\\
\infty\ (|r|>1)
\end{cases}
\end{eqnarray}
Sum of Geometric Sequences with Common Ratios as Fractions
Finally, I would like to explain what happens when the common ratio \(r\) is a fraction.
In conclusion, even if the common ratio \(r\) is a fraction, it does not change.
For example, suppose the first term \(a=2\), the common ratio \(r=\displaystyle \frac{1}{3}\), and the number of terms \(n=5\).
From the formula for the sum of geometric progressions,
\begin{eqnarray}
S_n&=&\displaystyle \frac{a(1-r^n)}{1-r}\\\\
&=&\displaystyle \frac{2\left( 1-\left( \displaystyle \frac{1}{3}\right)^5\right)}{1-\displaystyle \frac{1}{3}} \\
&=&\displaystyle \frac{2\left(1-\displaystyle \frac{1}{243} \right)}{\displaystyle \frac{2}{3}}\\
&=&\displaystyle \frac{2\left(\displaystyle \frac{242}{243} \right)}{\displaystyle \frac{2}{3}}\\
&=&\displaystyle \frac{2\times242\times3}{243\times2}\\
&=&\displaystyle \frac{242}{81}
\end{eqnarray}
Can be calculated.
The calculations themselves are a little more complicated, but they do the same thing!
That's all for the explanation of the sum of geometric progressions!
FAQs
How do you prove the sum of a geometric progression? ›
- To find the sum of finite (n) terms of a GP, Sn = a(rn - 1) / (r - 1) [OR] Sn = a(1 - rn) / (1 - r), if r ≠ 1. Sn = an, if r = 1.
- To find the sum of infinite terms of a GP, S = a / (1 - r), if |r| < 1 (and in this case, we say that the series converges)
The formula for the sum of an arithmetic sequence is: Sn=n2[2a+(n−1)d], S n = n 2 [ 2 a + ( n − 1 ) d ] , where: n = the number of terms to be added. a = the first term in the sequence. d = the constant value between terms.
How do you solve geometric progressions? ›The formula for the nth term of a geometric progression whose first term is a and common ratio is r is: an=arn-1. The sum of n terms in GP whose first term is a and the common ratio is r can be calculated using the formula: Sn = [a(1-rn)] / (1-r). The sum of infinite GP formula is given as: Sn = a/(1-r) where |r|<1.
How do you explain geometric proofs? ›Geometric proofs are a list of Statements and Reasons used to prove that a given mathematical concept or idea is true. Statements are claims about a geometric problem that cannot be proven true until backed by a mathematical Reason. Reasons are pieces of evidence that support a Statement.
What is the rule of sum of a geometric series? ›The sum of the terms of a geometric sequence. The sum of the first n terms of a geometric sequence, given by the formula: Sn=a1(1−rn)1−r, r≠1. An infinite geometric series where |r|<1 whose sum is given by the formula: S∞=a11−r.
What are 2 examples of geometric sequence? ›{2,6,18,54,162,486,1458,...} is a geometric sequence where each term is 3 times the previous term. Example 2: {12,−6,3,−32,34,−38,316,...}
How do you solve a sum and simplify? ›∆f (k) = f (b) - f (a). To simplify any finite sum whatsoever, all we need to do is find a function f such that ∆f is the function we're summing.
How do you find the nth term when given the sum? ›Step 1: The nth term of an arithmetic sequence is given by an = a + (n – 1)d. So, to find the nth term, substitute the given values a = 2 and d = 3 into the formula. Step 2: Now, to find the fifth term, substitute n = 5 into the equation for the nth term.
What are the 4 steps to solving geometry problems? ›- Determine what you need to calculate to solve the problem. ...
- Draw a diagram. ...
- Record all appropriate measurements. ...
- Pay attention to units.
To calculate the geometric mean of two numbers, you would multiply the numbers together and take the square root of the result.
What is the formula for geometric probability? ›
The formula of geometric probability is the expected area divided by the total area. We have Geometric Probability = Probable Area/Total Area.
What is the fastest way to memorize math formulas? ›- Develop a Keen Interest in the Subject.
- Employ Visual Memory.
- Try Different Learning Strategies.
- Get Rid of Distractions.
- Practice as Much as You can.
- Use Memory Techniques.
- Sleep on it.
- Understand the Formulas.
Write down the formula you want to memorize: Writing the formulas again and again will help you remember them for longer. Our brain has a tendency to remember what we write repeatedly. Therefore, writing down tough formulas will make it easier for you to recall them.
What are mnemonics for remembering formulas? ›For example, a popular mnemonic such as PEMDAS (Please/Parenthesis, Excuse/Exponents, My/Multiply, Dear/Division, Aunt/Addition, Sally/Subtractions) is an effective memory aid for evaluating expressions using the order of Operations.
What are the 3 types of proofs used in solving geometry problems and give example? ›Two-column, paragraph, and flowchart proofs are three of the most common geometric proofs. They each offer different ways of organizing reasons and statements so that each proof can be easily explained.
What are 3 types of proofs? ›There are many different ways to go about proving something, we'll discuss 3 methods: direct proof, proof by contradiction, proof by induction. We'll talk about what each of these proofs are, when and how they're used.
What are the 5 parts of a proof geometry? ›The most common form of explicit proof in high school geometry is a two column proof consists of five parts: the given, the proposition, the statement column, the reason column, and the diagram (if one is given).
What are the different types of geometric progression? ›Geometric progression can be divided into two types based on the number of terms it has. They are: Finite geometric progression (Finite GP) Infinite geometric progression (Infinite GP)
Is 2 4 6 8 a geometric sequence? ›2,4,6,8,10….is an arithmetic sequence with the common difference 2.
What are geometric patterns in math examples? ›Example 1: In the pattern 65, 64, 63, 62, 61, we are subtracting the consecutive numbers by 1 or each number gets decreased by 1. Each number is getting increased by 5. A sequence of numbers that are based on multiplication and division is known as a geometric pattern.
How do you explain sum to a child? ›
Students start with a closed fist and say “4”. Students then count up “5, 6, 7”, extending three fingers one at a time. Students now have three fingers extended, but remind them that the answer isn't 3. They started with a 4 in their fist and then counted up, so the answer is 7.
What is sum explain with example? ›The SUM function adds values. You can add individual values, cell references or ranges or a mix of all three. For example: =SUM(A2:A10) Adds the values in cells A2:10.
How do you find the sum of a number? ›We can obtain the sum of digits by adding the digits of a number. We can ignore the digit's place values to find the digit sum. For example, the digit sum of 185 is 1 + 8 + 5 or 14.
What is the best app for geometry answers? ›- Geometry solver ² - calculator. Education.
- GeoGebra Geometry. Education.
- Solving Pythagoras. Education.
- Triangle & Angle calculator. Education.
- GeometryMaster - Geometry. Education.
- Trigonometry Help Lite. Education.
Today's mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It's one of the seven Millennium Prize Problems, with $1 million reward for its solution.
What is the 4 basic rule in solving equation? ›We have 4 ways of solving one-step equations: Adding, Substracting, multiplication and division. If we add the same number to both sides of an equation, both sides will remain equal. If we subtract the same number from both sides of an equation, both sides will remain equal.
How is geometric mean used in real life? ›The geometric mean is also used for sets of numbers, where the values that are multiplied together are exponential. Examples of this phenomena include the interest rates that may be attached to any financial investments, or the statistical rates if human population growth.
What grade do you learn geometric mean? ›Lesson: Geometric Mean Mathematics • 10th Grade
In this lesson, we will learn how to find geometric means between two nonconsecutive terms of a geometric sequence.
Geometric probability is the calculation of the likelihood that you will hit a particular area of a figure. It is calculated by dividing the desired area by the total area. The result of a geometric probability calculation will always be a value between 0 and 1. If an event can never happen, the probability is 0.
What is an example of geometrical probability? ›The geometric probability is the area of the desired region (or in this case, not so desired), divided by the area of the total region. The area of the lava pit is 2 in × 10 in = 20 in2. The area of the screen is 10 in × 17 in = 170 in2. That makes the geometric probability or 11.8%.
What is the easiest trick to learn math? ›
Maths Division Tricks
If a number is an even number and ends in 0, 2, 4, 6 or 8, it is divided by 2. A number is divisible by 3 if the sum of the digits is divisible by 3. Consider the number 12 = 1 + 3 and 3 is divisible by 3. A number is divisible by 4 if the last two digits are divisible by 4.
What is dyscalculia? Dyscalculia is a learning disorder that affects a person's ability to understand number-based information and math. People who have dyscalculia struggle with numbers and math because their brains don't process math-related concepts like the brains of people without this disorder.
Who is the fastest person to do math? ›- Education.
- On International Maths Day 2022, meet world's fastest human calculator Neelakantha Bhanu Prakash.
Mnemonics
One way of doing this is by taking the first letters of a string of information you want to remember and then using them to create a more memorable phrase that you find easier to recall than the original information.
ROY G. BIV = colors of the spectrum (Red, Orange, Yellow, Green, Blue, Indigo, Violet.) This is by far the most popularly used mnemonic. To make an Expression or Word mnemonic, the first letter of each item in a list is arranged to form a phrase or word.
What are mnemonics give 5 examples? ›...
Peg Method Mnemonics
- one = bun.
- two = shoe.
- three= tree.
- four = door.
- five = hive.
- six = sticks.
- seven = heaven.
- eight = gate.
Acronyms are one of the most popular and widely used mnemonic strategies. Using this method, students memorize a single word in which each letter is associated with an important piece of information. This letter-association strategy is especially useful for remembering short lists of items or steps.
What is the sum of the terms in a sequence? ›The sum of the terms of a sequence is called a series .
What is a sequence and how do I find its terms and sums? ›An arithmetic sequence is a series of numbers in which each term increases by a constant amount. To sum the numbers in an arithmetic sequence, you can manually add up all of the numbers. This is impractical, however, when the sequence contains a large amount of numbers.
What is the fastest way of finding the sum of numbers and one or more columns? ›The quickest and easiest way to sum a range of cells is to use the Excel AutoSum button. It automatically enters an Excel SUM function in the selected cell. The SUM function totals one or more numbers in a range of cells.
What are examples of sequence in math? ›
An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression.
How do you find the number of terms in a sum? ›All you need to do is plug the given values into the formula tn = a + (n - 1) d and solve for n, which is the number of terms. Note that tn is the last number in the sequence, a is the first term in the sequence, and d is the common difference.
What is the formula of sequence series? ›b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d is the common difference.
How do you explain a sequence? ›A sequence is an ordered list of numbers . The three dots mean to continue forward in the pattern established. Each number in the sequence is called a term. In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on.
Which of the following shortcut keys can we use to find the sum of values? ›AutoSum is a fast, easy way to add up multiple values in Excel. You can access the AutoSum command from either the Home tab or the Formulas tab, but there is a keyboard shortcut that makes it even faster: Alt+=. To use this shortcut, simply hold down the Alt key, then press the equals sign on your keyboard.
Which key combination is used to get the sum of? ›The Autosum Excel Function[1] can be accessed by typing ALT + the = sign in a spreadsheet, and it will automatically create a formula to sum all the numbers in a continuous range.
What is 1 * 2 * 3 * 4 * 5 all the way to 100? ›Therefore, the sum 1 + 2 + 3 + 4 + 5 + . . . . . . + 100 = 5050 .