Classwork Exercise and Series (Mathematics-SS2): Geometric Progression And Mean

Geometric Series

You can take the sum of a finite number of terms of a geometric sequence. And, for reasons you’ll study in calculus, you can take the sum of an infinite geometric sequence, but only in the special circumstance that the common ratio r is between –1 and 1; that is, you have to have | r | < 1.

For a geometric sequence with first term a₁ = a and common ratio r, the sum of the first n terms is given by:

Note: Your book may have a slightly different form of the partial-sum formula above. For instance, the “a” may be multiplied through the numerator, the factors in the fraction might be reversed, or the summation may start at i = 0 and have a power of n + 1 on the numerator. All of these forms are equivalent. In the special case that | r | < 1, the infinite sum exists and has the following value:

S_n = a(rⁿ – 1)/ r – 1, this is applicable only when r  > 1. Where S_n is the sum of n terms of a GP.

Evaluate the following:

The first few terms are –6, 12, –24, so this is a geometric series with common ratio r = –2. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2). The first term of the sequence is a = –6. Plugging into the summation formula, I get:

(-6)[fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”no-repeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][1-(-2)²⁰ /1-(-2)] = (-6)(1-(1048576) / 1+2]

= (-6)[-61048575 / 3]

= (-2)(-1048575)

= 2097150

So the value of the summation is 2 097 150

Evaluate S10 for 250, 100, 40, 16,….

The notation “S₁₀” means that we need to find the sum of the first ten terms. The first term is a = 250. Dividing pairs of terms, we get 100 ÷ 250 = 2/5, 40 ÷ 100 = 2/5, etc, so the terms being added form a geometric sequence with common ratio r = 2/5. When we plug in the values of the first term and the common ratio, the summation formula gives me

S₁₀ = 250[1-(2/5)¹⁰ / 1- 2/5]

= 250[1 – 1024/9765625 / 3/5]

= 250(9764601/9765625)(5/3)

= 250/1 (3254867/1953125)

S₁₀ = 6509734/15625

Note: If you try to do the above computations in your calculator, it may very well return the decimal approximation of 416.62297… instead of the fractional (and exact) answer. As you can see in the screen-capture to the right, entering the values in fractional form and using the “convert to fraction” command still results in just a decimal approximation to the answer. But (warning!) the decimal approximation will almost certain be regarded as a “wrong” answer! Take the time to find the fractional form!

Geometric Mean

The geometric mean is NOT the arithmetic mean and it is NOT a simple average. It is the nth root of the product of n numbers. That means you multiply a bunch of numbers together, and then take the nth root, where n is the number of values you just multiplied. Did that make sense? Here’s a quick example:

Example:

What is the geometric mean of 2, 8 and 4?

Solution:

Multiply those numbers together. Then take the third root (cube root) because there are 3 numbers.

32 x 8 x 4 = 4

Naturally, the geometric mean can get very complicated. Here’s a mathematical definition of the geometric mean:

Remember that the capital ‘PI’ symbol means to multiply a series of numbers. That definition says to multiply k numbers and then take the kth root. One thing you should know is that the geometric mean only works with positive numbers. Negative numbers could result in imaginary results depending on how many negative numbers are in a set. Typically this isn’t a problem, because most uses of the geometric mean involve real data, such as the length of physical objects or the number of people responding to a survey.

Question

1. What is the 12^th term of the GP 2, 14, 98, ….?

A. 3(7¹²)  B. 2(7¹²)  C. 2(7¹¹)  D.  2(7¹²)

2. Find the sum of 10 terms of the GP 4, 8, 16, …, leave your answer in index form.

A. 2¹² – 2² B. 2¹² – 2³ C. 2¹³ – 2³ D. 2¹² – 2²

3. What is the geometric mean of 4, 9, 9, and 2?

A. 6.045 B. 5.045 C. 7.045 D. 6. 038

4. Find the Geometric Mean of 1, 2, 3, 4, 5.

A. 7  B. 2.605  C. 3.506  D. 4.405

5. What is the geometric mean of 3 and 12 is
A. 6  B. 7  C. 3  D. 7

Answers

1. C  2. D  3. B  4. B  5. A[/fusion_builder_column][/fusion_builder_row][/fusion_builder_container]