Arithmetic sequences and series
What is this formula used for?
tn = a + (n - 1)d
Finding the formula for the general term (tn)
a) Define "geometric series"
b) What formula is used for finding a geometric series?
a) The sum of a geometric sequence
b) Sn = [a(rn - 1)]/(r - 1)
Alex bought $7000 worth of rare coins. The coins appreciate in value by $200 a year. If he sells the coins for $11000, how many years ago did he buy them?
End of first year: t1
t1 = 7200
t2 = 7400
t3 = 7600
tn= a + (n - 1)d
11000 = 7200 + (n - 1)200
11000 = 7200 + 200n - 200
11000 = 7000 + 200n
4000 = 200n
20 = n
.: Alex bought the coins 20 years ago
Given the explicit formula, f(n) = 2n – 3, write the first 5 terms
f(1) = 2(1) – 3 f(2) = 2(2) – 3
= -1 = 1
f(3) = 2(3) – 3 f(4) = 2(4) – 3
= 3 = 5
f(5) = 2(5) – 3
= 7
How do you get consecutive rows of Pascal's Triangle?
By adding the two numbers above.
Ex.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Find the formula for the general term (tn) and for t22 for the sequence: 4, 9, 14,...
a = 4 d = 5
tn = 4 + (n - 1)5
= 4 + 5n - 5
= 5n - 1
t22 = 5(22) -1
=109
Determine the formula for the nth term, tn, and find t5 for the geometric sequence 54, 27, 13.5, ...
a = 54 r = 1/2
tn = arn - 1
tn formula: tn = 54(1/2)n - 1
t5 = 54(1/2)5 - 1
= 54(1/2)4
= 3.375
A certain bacteria has a population of 24 at the beginning of an experiment and doubles every minute. If the experiment lasted 2 hours, how many bacteria were there in the end?
a = 24 r = 2
2 hours = 60 minutes x 2 = 120
Sn = n/2 [(2a + (n - 1)d]
S120 = 120/2 [(2(24) + (120 - 1)2]
= 60[48 + 119(2)]
= 60(286)
= 17,160
.: There were 17,160 bacteria at the end of the experiment.
Write the first five terms
t1 = 4 tn = tn-1(3)
t2 = t2-1(3) t3 = t3-1(3) t4 = t4-1(3)
= 4(3) = 12(3) = 36(3)
= 12 = 36 = 108
t5 = t5-1(3)
= 108(3)
= 324
Using the sequence 5, 12, 19, 26, 33
a) Determine the finite difference
b) Determine the explicit formula for the nth term
n = term number tn = term value
tn: 5 12 19 26 33
n: 1 2 3 4 5
The finite difference is 7.
b) 7 must be in our formula since it is the common difference. If we multiply 7 by each term number, the result is 2 more than the tn . Hence, the explicit formula is:
tn = 7n - 2
Find the sum of the first 45 terms of the series
5 + 11 + 17 + ...
a = 5 d = 6 n = 45
Sn = n/2 [2a +(n - 1)d]
S45 = 45/2 [2(5) +(45 - 1)6]
= 22.5[10 +(44)6]
= 22.5(10 + 264)
= 22.5(274)
= 6165
Find S10 for the series 3 + 7 + 11 + …
S10 = [3(410 – 1)]/4 – 1
= [3(1048,576)]/3
=1048,576
In an arithmetic sequence t3 = 16 and t17= 128. Determine the first term, the common difference, and the sum of the first 20 terms.
d = (128-16)/(17-3) tn= a + (n - 1)d
= 112/14 16 = a + (3 - 1)8
= 8 16 = a + (2)8
16 = a + 16
0 = a
Sn = n/2 [2a +(n - 1)d]
S20 = 20/2 [2(0) +(20 - 1)8]
= 10[(19)8]
= 10(152)
= 1520
Expand using Pascal's Triangle
(3x + 9)5
1(3x)5(9)0 + 5(3x)4(9)1 + 10(3x)3(9)2 + 10(3x)2(9)3 + 5(3x)1(9)4 + 1(3x)0(9)5
= 243x5 + 3645x4 + 21870x3 + 65610x2 + 98415x + 59049
Expand using Pascal's Triangle
(7x–4)4
=1(7x)4(4)0 + 4(7x)3(4)1 – 6(7x)2(4)2 + 4(7x)1(4)3 – 1(7x)0(4)4
=2401x4 + 5488x3 – 4704x2 + 1792x – 256
How many terms are there in the sequence 1, 3, 6, ..., 57?
tn = a + (n - 1)d
57 = 1 + (n - 1)3
57 = 1 + 3n - 3
57 = 3n - 2
59 = 3n
19= n
.: There are 19 terms in the sequence
Find the number of terms in the geometric sequence
2, 6, 18, ..., 486
a = 2 r = 3
tn = arn - 1
486 = 2(3)n - 1
486/2 = (2/2)(3)n - 1
243 = 3n - 1
35 = 3n - 1
n - 1 = 5
n = 6
.: There are 6 terms in the sequence.
A new restaurant sold 10 four-course meals on their opening day. Each day after, the number of meals they sold increased by 5 from the day before.
a) Write the formula for the nth term of the sequence.
b) How many meals and how many courses were sold 1 week after opening?
t1 = 10 t2 = 15 t3 = 20 a = 10 d = 5
a) tn= a + (n - 1)d
tn= 10 + (n - 1)5
= 10 + 5n - 5
= 5 + 5n
b) Sn = n/2 [2a + (n - 1)d]
S7 = 7/2 [2(10) + (7 - 1)5]
= 3.5[20 + (6)5]
= 3.5(40)
= 140
140 meals x 4 courses = 560 courses
.: The restaurant sold 140 meals with a total of 560 courses.
Describe the pattern in this sequence and write the next 3 terms:
1, 1, 2, 3, 5, 8, 13, 21, ...
tn is the sum of the last two terms
34, 55, 89
Describe the pattern in this sequence and write the next three terms.
38, 8, 2, ...
The rate is 1/4.
0.5, 0.125, 0.03125
Find the sum of the arithmetic series given the first and last terms: a = 2 t10 = 74
d = (74 - 2)/(10 - 1)
= 72/9
= 8
Sn = n/2 [2a +(n - 1)d]
S10 = 10/2 [2(2) +(10 - 1)8]
= 5[4 + 9(8)]
= 5(76)
= 380
Find the sum of the series 2 + 8 + 32 + … + 512
r = 4 a = 2
tn = arn - 1
512 = 2(4)n - 1
512/4 = 2(4/4)n - 1
128 = 2n - 1
27 = 2n - 1
n - 1 = 7
n = 8
Sn = [a(rn - 1)]/r - 1
S8 = [2(48 - 1)]/4 - 1
= [2(65,535)]/3
= 43690.7
A record-breaking snowstorm in Yukon resulted in 40 cm of snow. However, the temperature rose and one-sixth of the snow melted daily. How much snow was left after 1 week?
If ⅙ is melting, ⅚ are remaining.
a = 40 x 5/6 = 33.3 d = 5/6
tn= arn – 1
t7= 33.3(5/6)7 – 1
t7= 33.3(5/6)6
t7= 11.2
.: There will be 11.2 cm of snow left after 1 week.
Determine a recursion formula for this sequence and find the 5th term
3, 8, 13, 18, ...
a=3 d=5
Since the common difference is 5, the nth term will always be the previous term + 5. Therefore, the formula is:
t1 = 3, tn = tn-1 + 5
t5= t5-1 + 5
= t4 + 5
= 18 + 5
= 23
Determine the formula for calculating numbers in Fibonacci's Sequence.
1, 1, 2, 3, 5, 8, 13, 21, ...
Since each term is the sum of the previous two terms, the formula must be:
tn = tn -1 + tn - 2