![]() The sum of the first 7 terms of the geometric sequence is We can now calculate the sum of the first seven terms of the Therefore, the first term of the sequence is 8 5 Now that we know the value of the common ratio, we can use the The first term of the geometric sequence is not equal to zero since the sumĪnd □ are nonzero, we can divide both sides of the Substituting these expressions into the first equation, To establish the first term and common ratio of the sequence and then findĮxample 5: Finding the Sum of n Terms of a Geometricįind the sum of the first 7 terms of a geometric sequence given We need to use our knowledge of geometric sequences in order Two nonconsecutive terms in a geometric sequence, and the sum of two different In our final example, we are given a multiplicative relationship between This means that the second term of the sequence isġ 0 6 × 2 = 2 1 2. Therefore, the first term of the sequence is 53 and the common ratio is 2. So let us substitute this into equation (5) The sum of all the terms of the geometric seriesģ 3 3 9 = □ ( 2 ) − 1 2 − 1 3 3 3 9 = □ ( 2 ) − □. ![]() We define the common ratio of a geometric sequence asġ 6 9 6 = □ ( 2 ) ( 2 ) 3 3 9 2 = □ ( 2 ). The last term is 1 696, and the common ratio is 2. Given information about some of its properties.Įxample 4: Finding the Geometric Sequence given Its Last Term,Ĭommon Ratio, and the Sum of All Its Termsįind the geometric sequence given the sum of all the terms is 3 339, In the fourth example, we need to find the geometric sequence, Last term is 1, and sum of all terms is 1 093 is 7. The number of terms of a geometric sequence whose first term is 729, We can rewrite the left-hand side of our equation such that Which we can substitute back into equation (3)ġ 7 2 9 1 3 = 1 3 1 3 1 3 = 1 3 . Next, we recall the formula for the sum of the first □ ![]() We can rewrite the right-hand side of our equation such that Let □ = 7 2 9.Īnd □ is the common ratio between terms: We define the first term of a geometric sequence as □. Term is 1, and sum of all terms is 1 093 is. Number of terms in a particular geometric sequence.Įxample 3: Finding the Number of Terms of a Finite Geometric Sequence given Its Sumįill in the blank: The number of terms of a geometric sequence whose first term is 729, last In our next example, we will need to rearrange our formulae to calculate the The sum of the geometric sequence 1 6, − 3 2, 6 4, …, 2 5 6 To find the sum of the series, we can now use the The formula for the □ t h term of a geometric The common ratio of the sequence is equal to − 2. We can calculate the value of the common ratio, □,īy dividing any term by the term that precedes it: Īlternatively, we could have subtracted (1)Įxample 2: Finding the Sum of a Finite Geometric Sequence įactoring □ from the right-hand side and If we multiply both sides of our equation by □, we have So the sum of the first □ terms of a geometric ![]() We will now derive a formula for the sum of the first □Ĭonsider a geometric sequence with first term □Īnd common ratio □. We can see that the sum of these terms is 59 048. In this case, by adding together the first 10 terms in the series, The sum of the terms in a sequence is called a series. Of multiplying the previous term by the common ration, we find that Since we multiply one term by the common ratio to get the next term,Īnd by dividing both sides of the equation byĪlternatively, with the definition that one term is the result
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