Arithmetic Sequences and Series
Arithmetic Sequences Exercise: Find the next term and the general formula for the following:
For each of these three sequences there is a common difference. In the first sequence the common difference is d = 3, in the second sequence the common difference is d = 4, and on the third sequence the common difference is d = 3. We will call a sequence an arithmetic sequence if there is a common difference. The general formula for an arithmetic sequence is
Example What is the difference between the fourth and the tenth terms of {2,6,10,14,...) We have a_{10}  a_{4} = (10  4)d = 6(4) = 24
Arithmetic Series First we see that 1+ 2 + 3 + ... + 100 = 101 + 101 + ... + 101 (50 times) = 101(50) In general
n(n + 1)
Example What is S = 1 + 4 + 7 + 10 + 13 +... + 46
Solution S = 1 + (1 + 1(3)) + (1 + 2(3)) + (1 + 3(3)) + ... + (1 + 15(3)) = (1 + 1 + ... + 1) + 3(1 + 2 + 3 + ... + 15) = 16 + 3(15)(16)/2 In General
d(n  1)(n) = 1/2 [2n(a_{1}) + d(n  1)(n)] = 1/2[2n(a_{1})+ dn^{2}  dn] = (n/2)[2(a_{1})+ dn  d]= (n/2)[2(a_{1}) + d(n  1)]
Or Alternatively
Example How much will I receive over my 35 year career if my starting salary is $40,000, and I receive a 1,000 salary raise for each year I work here? Solution We have the series: 40,000 + 41,000 + 42,000 + ... + 74,000 = 35/2 (40,000 + 74,000) = $1,995,500 For an interactive lesson on how to determine a term of an arithmetic sequence given two other terms Click Here
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