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Infinite Geometric Series Formula
The sum of an infinite geometric sequence yields an Infinite Geometric Series Formula. In this series, there would be no final term. The infinite geometric series has the formula as a1+a1r+a1r2+a1r3+…, where a1 is the first term and r is the common ratio.
When the common ratio in an infinite geometric series is greater than one, the terms in the Infinite Geometric Series Formula grow larger and larger and adding the larger integers yields no final answer. Infinity is the only feasible answer. As a result, one does not consider the common ratio bigger than one for an infinite geometric series.
What Is Infinite Geometric Series Formula?
The Infinite Geometric Series Formula is similar to the arithmetic sequence, except that each term is multiplied by an additional factor of r. The r’s exponent will be one less than the term number. The first term has never been multiplied by r. (exponent on r is 0). The second term has already been multiplied by r. The third word has twice been multiplied by r, and so on.
Another type of geometric series is the Infinite Geometric Series Formula. The sum of an infinite geometric sequence yields an infinite geometric series.
When the ratio is greater than one, the terms in the Infinite Geometric Series Formula grow larger and larger, and if one adds larger and larger integers indefinitely, they will receive infinity as an answer. When the magnitude of the ratio is bigger than one, one does not deal with infinite geometric series.
The magnitude of the ratio cannot be one because the series would no longer be geometric and the sum formula would have a division by zero.
A geometric series can be stated as the sum of the first term and a multiple of the common ratio since each term is a common multiple of the term that came before it. If a1 is the first term, an is the nth term, and r is any geometric sequence’s common ratio, then an = a1 rn1.
A geometric series is simply a geometric sequence that has been multiplied by itself. akin to this
1 + 2 + 4 + 8 + 16 + 32 + …
It is written as the sum of all the terms rather than simply listing them with commas between them.
The same sigma notation is applied here as well like in the arithmetic series. The numbers in this series will begin at n = 1 and continue all the way to infinity. In this way, the geometric series is special.
The sequence terms will grow quite huge if the common ratio r is more than one. Adding excessively huge numbers makes it nearly hard to obtain the outcome. The solution in this situation can be regarded as endless. It is possible to determine the common ratio’s total when its value ranges between 1 and 1. Therefore, it can be claimed that r<1 is a legitimate value for the sum of an infinite geometric series. S can be used to symbolise the sum. As the sequence of the sum converges closer to a specific value, an infinite geometric series with a fixed sum is referred to as a convergent series. If r > 1, the series is divergent, and since the numbers keep becoming bigger and bigger in a sequential manner, the sum will eventually be infinite (and thus there will be no definitive sum). Students can use the sequence “10 + 20 + 40 + 80 +…” as an illustration. The common ratio in this situation would be 2. As they can see, the values keep increasing as the series goes on indefinitely, making it impossible for them to reach a certain sum.
Examples using Geometric Infinite Series Formula
Students are advised to keep revising the theories of topics. All the questions can be solved well after complete revision. Students are advised to practice examples related to the Infinite Geometric Series Formula. If students are stuck while solving questions related to the Infinite Geometric Series Formula they can make use of the solutions available on Extramarks. The NCERT solutions provided by Extramarks are beneficial for solving questions regarding the Infinite Geometric Series Formula.
FAQs (Frequently Asked Questions)
1. What is the Infinite Geometric Series Formula?
The Infinite Geometric Series Formula is represented as a1+a1r+a1r2+a1r3+…, where a1 is the first term and r is the common ratio. The Infinite Geometric Series Formula is a very important concept.
2. Which is the best place to access NCERT Solutions for Mathematics?
The Extramarks website and mobile application have authentic NCERT solutions that students can use to solve exercises in each chapter. Extramarks has multiple practice materials that students can make use of to score well in their upcoming examinations.