By M Willatt
1 When given a sequence, you first need to check if it’s ARITHMETIC. Remember arithmetic is when you are adding or subtracting the same number to get to the next term. Examples: 0,11, 22, 33, 44, 55, 66, …
d = 11
-5, -5.5, -6, -6.5, -7, -7.5,…
d = -.5
54, 46, 38, 30, 22, 14, 6,…
d = -8
1, 1.1, 1.2, 1.3, 1.4, 1.5, … 1/10
d=
2 If it’s not arithmetic, next check if it’s GEOMETRIC. Remember geometric is when you multiply by the same number to get to the next term. *Note: Multiplying by a fraction can be the same as dividing… terms can get smaller. Examples: 7, 21, 63, 189, …
r=3
1, -1/2, 1/4, -1/8,…
r = -1/2
-5, -50, -500, -5000,…
r = 10
4, 1, 1/4, 1/16, 1/64 …
r = 1/4
3
If it’s not arithmetic or geometric, you would say it is NEITHER. It is still a sequence… just not one we have a specific formula for. You could still write one, but you usually wouldn’t be asked to do that. Examples: 1, 13, 26, 40,… We are adding 12, then 13, then 14, and so on. It is NOT arithmetic because we are adding a different number each time. 34, 304, 3004, 30004,… We are multiplying 3 by multiples of 10 and then adding 4. 5, 12, 2, 9, -1, 6,… This time we’re adding 7 and next subtracting 10.
Below are the formulas you’ll be given on the final exam. It seems easy enough, but during the test you will need to know which one to use.
tial r a p a d fin st “n” e, o t l ir Use ...the f examp sum s. For sum of term u need erms. t if yo irst 20 f the
Use to fin examp d a specific t erm. F le, or term, y if you need t he 15 th ou wou ld sub a 15 in for n
Notice ther e is no “n” because yo adding ALL u’re the terms n o t just the first “n” term s This only works because it converges when |r|<1. Formula sheet from www.ncpublicschools.org/docs/accountability/common-exams/formula-sheet.pdf CC0
.
Once you know which one to use, remember you also need to know what the variables mean...
Arithmetic Partial Sum:
Do you know what to substitute in for the variable for geometric problems?
Geometric Partial Sum:
Geometric Infinite Sum:
Now that we’ve reviewed the formulas, let’s try some problems. a)State if the following sequences are arithmetic, geometric, or neither. b)Then tell whether they converge or diverge.
1. 3, 6, 9, 12, 12,… a) b)
arithmetic adding 3 each time diverge because arithmetic always diverge
2. 3, 9, 27, 81,… • •
geometric multiplying by 3 each time diverge because r > 1
• 425, 85, 17, 3.4,… a) geometric multiplying by 1/5 each time b) converge because r < 1
a) 3000, 1500, 500, 125, 25,… a) neither multiplying by 1/2, then 1/3, then 1/4, … b) converge the denominator of the multiplier is getting larger meaning the terms keep getting smaller (and will eventually be very close to 0); however, we don’t have a formula for this infinite sum as it is NOT geometric
Write the explicit formula for the sequences below:
1. 1, 7/2, 6, 17/2, 11,… First, let’s determine what type of sequence this is. To check for arithmetic subtract to find the difference: 7/2 – 1 = 5/2, 6 – 7/2 = 5/2, 17/2 – 6 = 5/2 It IS arithmetic since we are adding 5/2 each time so we’ll use an= a1 + (n - 1)d.
an= 1 + (n - 1)(5/2)
2. 160, 120, 90, 67.5,… First, let’s determine what type of sequence this is. To check for arithmetic subtract to find the difference: 120 – 160 = -40, 90 – 120 = -30, 67.5 – 90 = -22.5 The difference is NOT the same so it is NOT arithmetic. Next, let’s check for geometric. To check, you divide a term by the one before it to find the ratio: 120 ÷ 160 = 3/4, 90 ÷ 120 = 3/4, 67.5 ÷ 90 = 3/4 It IS geometric since we are multiplying by 3/4 each time so we’ll use an= a1(r)n-1.
an= 160(3/4)n-1