A geometric series is a series whose related sequence is geometric. It results from adding the terms of a geometric sequence .
Example 1:
Finite geometric sequence:
Related finite geometric series:
Written in sigma notation:
Example 2:
Infinite geometric sequence:
Related infinite geometric series:
Written in sigma notation:
Finite Geometric Series
To find the sum of a finite geometric series, use the formula,
,
where is the number of terms, is the first term and is the common ratio .
Example 3:
Find the sum of the first terms of the geometric series if and .
Example 4:
Find , the tenth partial sum of the infinite geometric series .
First, find .
Now, find the sum:
Example 5:
Evaluate.
(You are finding for the series , whose common ratio is .)
Infinite Geometric Series
To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, ,
where is the first term and is the common ratio.
Example 6:
Find the sum of the infinite geometric series
.
First find :
Then find the sum:
Example 7:
Find the sum of the infinite geometric series
if it exists.
First find :
Since is not less than one, the series does not converge. That is, it has no sum.
FAQs
The formula for the sum of a finite geometric series of the form a+ar+ar^2+... +ar^n is given by S = a(1-r^(n+1))/(1-r). This formula can be obtained by setting S = a+ar+ar^2+... +ar^n, multiplying both sides by -r, then adding the two formulas and simplifying.
How to find the limit of a geometric series? ›
When the absolute value of the common ratio (r) is between 0 and 1, the limit of the series converges to a finite sum. The formula for the sum is a / (1 - r), where a is the first term.
What is the rule for geometric series? ›
A geometric series is a unit series (the series sum converges to one) if and only if |r| < 1 and a + r = 1 (equivalent to the more familiar form S = a / (1 - r) = 1 when |r| < 1).
How to solve geometry problems easily? ›
This approach is similar to that for solving almost a word problem, but is geared slightly more toward the characteristics of geometry problems in particular.
- Determine what you need to calculate to solve the problem. ...
- Draw a diagram. ...
- Record all appropriate measurements. ...
- Pay attention to units.
Each term of a geometric sequence is formed by multiplying the previous term by a constant number r, starting from the first term a1. Therefore, the rule for the terms of a geometric sequence is an=a1(r)^(n-1).
What are the two formulas of geometric series? ›
The formulas for geometric series with 'n' terms and the first term 'a' are given as, Formula for nth term: nth term = a r. Sum of n terms = a (1 - rn) / (1 - r) Sum of infinite geometric series = a / (1 - r)
How do you solve geometric sequence problems? ›
To continue a geometric sequence:
- To identify r, take two consecutive terms from the sequence.
- Divide the second term by the first term to find the common ratio. r . \textbf{r}. r.
- Multiply the last term in the sequence by the common ratio to find the next term.
- Repeat Step 3 for each new term.
How do I prove the geometric series formula?
- Write out the sum once.
- Write out the sum again but multiply each term by r.
- Subtract the second sum from the first. All the terms except two should cancel out.
- Factorise and rearrange to make S the subject.
The general formula for finding the sum of an infinite geometric series is s = a1⁄1-r, where s is the sum, a1 is the first term of the series, and r is the common ratio. To find the common ratio, use the formula: a2⁄a1, where a2 is the second term in the series and a1 is the first term in the series.
What is the formula for the sum 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.
A finite geometric series can be solved using the formula a(1-rⁿ)/(1-r). Sal demonstrates how to derive a formula for the sum of the first 'n' terms of such a series, emphasizing the importance of understanding the number of terms being summed.
How to solve geometric progression? ›
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).
What is the explicit formula for the geometric sequence? ›
Explicit Geometric Sequence Formula: The explicit formula for a geometric sequence is a n = a 1 r n − 1 , where is the first term of the sequence, is the common ratio, and is the integer index of the terms of the sequence.
What is the general rule formula for the geometric sequence? ›
The general form of the geometric sequence formula is: an=a1r(n−1), where r is the common ratio, a1 is the first term, and n is the placement of the term in the sequence. Here is a geometric sequence: 1,3,9,27,81,…
How to solve terms of a geometric sequence? ›
In geometric sequences, we use multiplication to find each subsequent term. The number we multiply by is called the common ratio. Each term gets multiplied by the common ratio, resulting in the next term in the sequence.
How do you find the missing number in a geometric sequence? ›
Step 1: Find the common ratio of each pair of consecutive terms in the sequence by dividing each term by the term that came before it. Step 2: Multiply the common ratio with the number prior to the first missing number in the sequence. Step 3: Repeat Step 2 for any other missing numbers.
