In a geometric progression, each successive term is obtained by multiplying the common ratio to its preceding term.Important Notes on Geometric Progression: The variation of the terms is non-linear. Here are a few differences between geometric progression and arithmetic progression shown in the table below: Geometric ProgressionĪP has the same common difference throughout.Ī new term is the product of the previous term and the common ratioĪ new term is the sum of the previous term and the common difference.Īn infinite geometric progression is either divergent or convergent.Īn infinite arithmetic progression is always divergent. Geometric Progression vs Arithmetic Progression But when |r| ≥ 1, then the terms become larger and larger infinitely and hence we cannot determine the sum in this case. This is because when the common ratio is less than 1 (a proper fraction), the terms become smaller and smaller as we go forward and they are equivalent to 0. Subtracting equation (2) from equation (1), Proof of Sum of Infinite Geometric Progression FormulaĬonsider an infinite geometric sequence a, ar, ar 2. and the sum of the first n terms, in this case, S n = a + a + a +. If r = 1, the progression looks like a, a, a. Since (r - 1) is in its denominator, it is defined only when r ≠ 1. Subtracting equation (1) from equation (2), Proof of Sum of Finite Geometric Progression FormulaĬonsider a finite geometric progression of n terms, a, ar, ar 2. If the number of terms in a geometric progression is infinite, then the sum of the geometric series is calculated by the formula: If the number of terms in a geometric progression is finite, then the sum of the geometric series is calculated by the formula: As we read in the above section that geometric progression is of two types, finite and infinite geometric progressions, hence the sum of their terms is also calculated by different formulas. The geometric progression sum formula is used to find the sum of all the terms in a geometric progression. is an infinite series where the last term is not defined. It is the progression where the last term is not defined. Infinite geometric progression contains an infinite number of terms. It is the progression where the last term is defined. Finite geometric progressionįinite geometric progression contains a finite number of terms. Let us see the information about each of these. The geometric progression is of two types. To find the terms of a geometric series, we only need the first term and the constant ratio. The common ratio can have both negative as well as positive values. where 'a' is the first term and 'r' is the common ratio of the progression. The GP is generally represented in form a, ar, ar 2. At the end of the first year you will have a total of: \ With simple interest, the key assumption is that you withdraw the interest from the bank as soon as it is paid and deposit it into a separate bank account.A geometric progression is a special type of progression where the successive terms bear a constant ratio known as a common ratio. You are paid $15\%$ interest on your deposit at the end of each year (per annum). We refer to $£A$ as the principal balance. Simple and Compound Interest Simple Interest For example, \ so the sequence is neither arithmetic nor geometric. A series does not have to be the sum of all the terms in a sequence. The starting index is written underneath and the final index above, and the sequence to be summed is written on the right. We call the sum of the terms in a sequence a series. The Summation Operator, $\sum$, is used to denote the sum of a sequence. If the dots have nothing after them, the sequence is infinite. If the dots are followed by a final number, the sequence is finite. Note: The 'three dots' notation stands in for missing terms. is a finite sequence whose end value is $19$.Īn infinite sequence is a sequence in which the terms go on forever, for example $2, 5, 8, \dotso$. For example, $1, 3, 5, 7, 9$ is a sequence of odd numbers.Ī finite sequence is a sequence which ends. Contents Toggle Main Menu 1 Sequences 2 The Summation Operator 3 Rules of the Summation Operator 3.1 Constant Rule 3.2 Constant Multiple Rule 3.3 The Sum of Sequences Rule 3.4 Worked Examples 4 Arithmetic sequence 4.1 Worked Examples 5 Geometric Sequence 6 A Special Case of the Geometric Progression 6.1 Worked Examples 7 Arithmetic or Geometric? 7.1 Arithmetic? 7.2 Geometric? 8 Simple and Compound Interest 8.1 Simple Interest 8.2 Compound Interest 8.3 Worked Examples 9 Video Examples 10 Test Yourself 11 External Resources SequencesĪ sequence is a list of numbers which are written in a particular order.
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