Sigma notation is used to denote summations in a compact format.

This is the nth partial sum of the sequence an, written in sigma notation

Geometric Series Multiply

This lesson will concentrate on series related to geometric sequences, called geometric series. We will be working with finite sums (the sum of a specific number of terms).

Geometric Series: S_n = a₁ +  (a₁r) +  (a₁r²) + (a₁r³) +  (a₁r⁴) +  ... + (a₁r^{n - 1})

A geometric series is the adding together of the terms of a geometric sequence.

Formulas used with geometric sequences and geometric series:

To find any term of a geometric sequence: where a1 is the first term of the sequence, r is the common ratio, n is the number of the term to find. Note: a1 may be simply referred to as a.

To find the sum of a certain number of terms of a geometric sequence: where Sn is the sum of n terms (nth partial sum), a1 is the first term, r is the common ratio, r not equal to one. Could also be written as Could also be written as

Click here to see "How these Formulas were Created".

Let's take a look at a variety of examples working with geometric sequences and series.
Read the "Answers" carefully to find "hints" as to how to deal with these questions.

Questions:	Answers:
1. Find the value of .	First, notice that the index starts with 0. This is geometric, common ratio 2, so the series formula could be used. Notice that the number of terms is 3. But, sometimes the sigma has only a few terms and the sum can be found by mental addition. Unless a formula is specifically requested in the question, you can just add.
2. A geometric sequence is defined by the recursive formula a1 = 6 and an = 3 • an - 1. Find the value of	The sigma indicates that the sum will be the addition of 6 terms from k = 1 to k = 6. Use the recursive formula to find the terms and add. 6 + 18 + 54 + 162 + 486 + 1458 = 2184 OR Since the formula is geometric and the question is asking for the sum of the first 6 terms, go right to the series formula ( r = 3 ).
3. Evaluate using a series formula:	Which series is needed: arithmetic or geometric? List the terms from the summation to get a better idea of what is happening in the series. This is clearly a geometric series with a common ratio of 3. By observation: n = 5, a1 = 3, r = 3 Be sure to show use of the geometric formula.
4. Use the geometric series formula to determine the value of	Be careful on the number of terms, n. The index goes from 0 to 9 which means there will be 10 terms (not 9). a1 = 10, r = 2, n = 10
5. Find the sum of the first 8 terms of the sequence: -5, 15, -45, 135, -405, ...	First determine that the sequence is geometric. It is, since each term is being multiplied by -3. By observation: n = 8, a1 = -5, r = -3 Notice the word SUM. Use the series formula.
6. Find the sum of the first 5 terms of a geometric sequence whose first term 24 and whose common ratio is ½.	Grab the geometric series formula: Yes, there is a decimal in this answer. The sequence is {24, 12, 6, 3, 1½, ...}
7. Find the sum of a geometric sequence whose first term is 40, the last term is 3906.25, and the number of terms is 6.   Notice how BOTH formulas work together to arrive at the answer.	We need to know the ratio, r. Use the sequence formula to find r, then use the series formula to find the sum.
8. The second and third terms of a geometric sequence are 8 and 16. Find the sum of the first 20 terms of this sequence.	We know the sequence is geometric, so the common ratio to get from 8 to 16 is r = 2. To get the first term, work backward. If the second term is 8, the first term must be 4, so that 4 • 2 = 8. The question asks for the sum:
9. The second term of a geometric sequence is 9 and the fourth term is 81. Find the sum of the first 10 terms of this sequence.	The word "sum" indicates the need for the series formula. Both formulas will be needed. We are missing the value of r and a1for the sum formula. Get a visual of what is given. ___, 9, ___, 81, ... Let a1 = 9, a3= 81 (temporarily) and find r. We are still missing a1 for the sum formula, so work backward (dividing 9 by 3) to get the first term of 3. Now, we use this information to find the sum:
10. Find the sum of the sequence { 4, 8, 16, ..., 1024}         Remember that geometric series are exponential functions, so keep those exponential equation solving skills in mind.	We don't know the number of terms, n, but we do know the first and last terms. Use the sequence formula for an to find the number of terms.. The sequence is geometric, common ratio r = 2. Now find the sum.
11. Russian nesting dolls are a set of wooden dolls of decreasing size placed inside one another. In order to fit properly, each doll must be 4/5 the height of the previous doll. The tallest doll is 10 inches in height. Using geometric sequences and series, determine the answers to the following questions: a) What is the height of the shortest doll in a set containing 5 dolls? b) What is the combined height of all five dolls in the set?	a) In a geometric sequence of heights, the shortest doll will be the fifth term. The first term is 10, the r = 4/5. b) A geometric series is needed for the combined heights.