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Introduction to Sum of terms of Geometric Progression
A geometric progression (also known as a geometric sequence) is a numerical series in which each term is created by multiplying the preceding term by a constant factor known as the common ratio. The sum of terms in a geometric progression is a fundamental notion that allows us to compute the total value produced by adding all of the elements in the sequence. This article investigates the formula for calculating the sum of terms, its restrictions, convergence, and examples, as well as commonly asked issues and solutions.
What is sum of GP
The formula for calculating the sum of terms in a finite or infinite geometric progression is different.
- Formula for sum of n terms of GP
Sum of a Finite Geometric Progression: The sum (denoted by “S”) of a geometric progression with “n” terms may be determined using the formula:
S = a(rn – 1)/r – 1
In this case, “a” stands for the first term, “r” stands for the common ratio, and “n” stands for the number of terms.
Formula for infinite GP
- Sum of an Infinite Geometric Progression: The sum (denoted by “S”) of an infinite geometric progression (where the common ratio “r” is between -1 and 1) can be computed using the formula:
- S = a/1 – r. The collection of input values for which the function is defined is referred to as the domain.
Limitations and convergence of sum of GP
- Limitations: The sum of a finite geometric progression formula is only valid when the number of terms is known.
- Only when the common ratio is between -1 and 1 can the formula for the sum of an infinite geometric progression be used.
- An infinite geometric progression converges if the common ratio’s absolute value is less than one (|r| 1).
- An infinite geometric progression’s total converges to a finite value when |r| 1.
- If |r| is less than 1, the total approaches infinity or oscillates without a fixed value, which is known as the infinite geometric progression diverging.
Conclusion
The sum of terms in a geometric progression is important in many mathematical applications, such as physics, engineering, and finance. We can compute the sum of terms precisely and interpret the results by comprehending the formulas, restrictions, and convergence features. The formula offers a potent tool to assess the cumulative value generated by adding the terms in the sequence, regardless of whether it is a finite or infinite geometric progression.
Problems related sum of GP
- Finite geometric progression illustration
- Consider the geometric progression where the initial term (a), common ratio (r), and number of terms (n) are all equal to 2, 3, and 4. Using the following formula to calculate the total of a finite geometric progression:
S = 2 * (3^4 – 1) / (3 – 1)
S = 2 * (81 – 1) / 2 S = 80
- Infinite geometric progression illustration Think about the geometric progression where the common ratio (r) is 1/2 and the first term (a) is 1. Solution: Using the sum of an infinite geometric progression formula:
S = 1 / (1 – 1/2)
S= 1 / (1/2)
S = 2
Frequently asked questions Sum of GP
What is the geometric progression first n terms total value?
The formula S = a * (1 - rn) / (1 - r) can be used to compute the sum of the first 'n' terms.
A geometric path that is infinitely long can either converge or diverge, so how can we tell?
If the common ratio (|r|) has an absolute value smaller than 1, an infinite geometric progression converges. Otherwise, it turns off.
When a geometric progression is endless, may its sum be negative?
If the common ratio is negative and the terms in the series alternate signs, the sum can indeed be negative.
What occurs if a geometric progression's common ratio is equal to 1?
All terms in the sequence will be identical if the common ratio is 1, and the sum will be the product of the common ratio and the number of terms.
What is the sum of geometric progression formula for infinite terms
S = a / (1 - r), where 'a' denotes the sequence's initial term and 'r' denotes its common ratio, is the formula for an infinite geometric progression's sum. With the help of an infinite number of terms in the sequence, this formula calculates the total value.
What is the sum of n terms of geometric progression when r =1.
The formula S = n * a, where 'n' is the number of terms and 'a' is the first term, can be used to calculate the sum of 'n' terms in a geometric progression when the common ratio (r) is equal to 1. The sum in this instance is only the first term multiplied by the number of terms.
When does the sum of infinite GP converge:
When the common ratio (|r|) is smaller than 1, the sum of an infinite geometric progression converges. In these scenarios, the sum gets closer to a limited value. If the common ratio's absolute value is larger than or equal to 1, the series diverges and does not have a finite total.
What is the sum of first n terms in Geometric Sequence
The formula for calculating the sum of the first 'n' terms in a geometric sequence is S = a * (1 - rn) / (1 - r), where 'a' stands for the first term and 'r' is the common ratio. The cumulative value obtained by adding the stated number of terms in the sequence is calculated using this equation.
What is the sum of the infinite GP when the common ratio is
The total of the infinite series depends on the value of the first term (a) when the common ratio (r) in a geometric progression equals 1. The series diverges and does not have a finite sum if an is less than 0. The sum of all terms is 0 if a = 0.
