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Math 12 Pre-calculus
Arithmetic sequences
An arithmetic sequence is a pattern of numbers where successive terms have a common difference – that is, tn – tn-1 = d. An
arithmetic sequence is the discrete version of a continuous linear function.
Some recursive sequences are also arithmetic sequences.
The nth term (the general term) of a sequence can be found using
tn  t1  (n  1)d
This is sometimes written in an expanded form
where
1.
t1 is the first term
n is the term number (the index)
d is the common difference between terms
Which of these sequences are arithmetic?
b) tn = tn-1 – 3 , t1 = 10
c) tn = n2
a) 7, -17, -41, -65….
c) tn = tn-2 – tn-1 , t1 = 4 , t2 = 7
2.
Write the general formula (the discrete function tn) for each arithmetic sequence. Write it in the form tn = t1 + (n-1)d,
then expand your function.
a) 12, 15, 18, 21…
c) tn = tn-1 + 6 , t1 = -2
e) 1, 2.5, 4, 5.5…
3.
Thinking back to last year: calculate finite differences (1st differences, 2nd differences, 3rd differences, etc) to determine
the type of function (n, n2, n3, etc) that models each sequence.
a)
c)
4.
b) 7, 3, -1, -5…
d) The set of positive integers divisible by 3
2, 11, 34, 77, 146, …
-3, -5, 17, 111, 349, 827, 1665, …
b) 3,5,9,15, 23, …
d) 3, 2, -3, -12, -25, …
Find the 50th term for each of the arithmetic sequences described below
a) If the 3rd term is 20, the 6th term is 38….
b) If the 2nd term is -8 and the 11th term is 21.5…
c) If the 99th term is 1450 and the 100th term is 1700…
5.
If an arithmetic sequence contains…
a) A 1st term of 8 and a 4th term of 12, does this sequence contain a term of 866? If so, what term?
b) A 4th term is 15 and the 20th term is 79, does the sequence contain the term 291?
c) A 9th term of -16 and a 25th term of 21, does this sequence contain a term of 1000? If so, what term?
6.
An arithmetic mean is the same as a numerical average of two numbers. i.e. the arithmetic mean of 10 and 6 is 8.
Another way of interpreting an arithmetic mean is the number that forms an arithmetic sequence with those around it.
Notice that 6,8,10 form an arithmetic sequence. Use this fact to insert the number of arithmetic means specified for the
conditions given:
a) insert an arithmetic mean between 12 and 20
b) insert two arithmetic means between 7 and 145
c) insert five arithmetic means between 10 and 22
7.
From your textbook:
Page 3 - #2
Page 5 - #5, #6a,b, #7iii
Page 6 - #13
Page 7 - #14, #17a