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Integrated math 3

Course: Integrated math 3>Unit 2

Lesson 7: Geometric series

Geometric series introduction

A geometric series is the sum of the first few terms of a geometric sequence. For example, 1, 2, 4, 8,... is a geometric sequence, and 1+2+4+8+... is a geometric series. See an example where a geometric series helps us describe a savings account balance.

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• I still cannot get it. Why do we get 1000 + 1000(1.05) which is 1000 + 1050 = 2050 on the first year; I think we should get 1000 + 1000(0.05) or 1000(1.05).

I would appreciate any help, thank You all for this project by the way.
• Every year he is depositing \$1000. However, the \$1000 deposited in previous years is still earning interest.
For example, in the first year, he deposits \$1000.
In the second year, he gains 5% interest on the \$1000, and now it becomes \$1050. But since he is depositing \$1000 every year, in the second year he has a total of \$2050, since he has not yet earned interest on the second \$1000 that he deposited. In the third year the first \$1000 that he deposited that had turned into \$1050, gains %5 interest and is again multiplied by 1.05. basically, the first hundred now has been multiplied by 1.05 twice. 1000(1.05)(1.05) or 1000(1.05)^2. The second \$1000 he deposited gains %5 interest and turns into 1000(1.05)^1 and then since it is the beginning of the year he deposits one more \$1000, which has not gained interest yet, so is 1000(1.05)^0 or just 1000.
So the pattern in the third year is:
1000 + 1000(1.05)^1 + 1000(1.05)^2 + 1000(1.05)^2.
Note that the highest exponent is 2 which is one less than the year number. So Khan is correct. You forgot that every year he deposits \$1000 and the money from the previous years collect more and more interest as the years go by.
• In the description above it reads: "For example, 1, 2, 4, 8,... is a geometric sequence"...

I understand the process of the sequence but I'm more looking for the history of the terminology so to say. Geometry is about (measuring) shapes and sizes. Why is this multiplying sequence not called a 'multiplying sequence', or a mathimatical sequence or what have you. Of all the possible branches of maths it had to be geometry. Why?
• It kinda just happened that way...

The ancient Greek mathematician Euclid first wrote about these types of sequences in his book Elements. Because so much of Euclid's Elements deals with geometry, these sequences ended up being called geometric sequences (even though they aren't technically geometric).

So the "geometric" label is an historical accident, but kind of interesting if you know the story.

Hope this helps!
• Why is this geometric series stuff part of the Factoring Polynomials unit?
• agreed lol, I think it's fascinating but a very jarring transition
• Multiplying the balance by the 1.05 is not 5% growth per year, it is 105% growth. I think it must be:
1000 + 1000(0.05) or 1000(1.05)
• 1.05 is just 5 percent growth plus the original if you just wanted the growth value you would do balance times .05 . That would be just growth
• So a question about sequences will say " What's the deposit value of the of the 8th day? " but about series would say " How much money does they have on the 8th day ? " Am I right ?
• Yes, that does seem correct. One asks for a specific term in the sequence, while the other asks for the sum of the terms.
• What is the difference between a geometric sequence and an exponential function? They seem to be the same thing, so I am confused.
• A sequence is specifically gotten from a list of numbers, but can be transformed into an exponential function. Also since geometric sequences always have the same starting point there will be no horizontal or vertical shifts in the function.
• Would 1000+(1000(1.05)^n-1)^n work?
• I don't think the ^n at the end would work because you want to add each year's deposit including their accrued interest plus the 1000, not multiplying by itself. I don't know what the formula would be though, he never finished teaching it.