Section 11.2 Arithmetic Sequences. Objectives: Recognizing arithmetic sequences by their formulas Finding the first term, the common difference, and find partial sums. Definition of Arithmetic Sequence.
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Recognizing arithmetic sequences by their formulas
Finding the first term, the common difference, and find partial sums.
An arithmetic sequence is in which the difference between each term and the preceding term is always constant.
For example: 2,4,6,8,… is an arithmetic sequence because the difference between each term is 2.
1,4,9,16,25,… is not arithmetic because the difference between each term is not the same.
Yes, the sequence is arithmetic because the difference between each term is constant. The common difference is -4
No, the sequence is not arithmetic because the difference between each term in not the same.
The formula for the nth term of an arithmetic sequence is
an =a1 + (n – 1)d
a) Find the first 5 terms of the sequence
The first 5 terms are 3, 4.5, 6, 7.5, and 9
b) Find the common difference, d.
d = 1.5
Since a1=13 and a2=7 then the common difference is 7 - 13 = -6.
So the nth term is
an =a1 + (n – 1)d
an = 13 – 6(n – 1)
13, 7, 1, –5, –11, –17, . . .
a300 = 13 – 6(299) = –1781
To find the nth term of the sequence, we need to find a and d in the formula an= a + (n – 1)d
From the formula, we get: a11 = a + (11 – 1)d =a + 10d a19 = a + (19 – 1)d =a + 18d
For the arithmetic sequence an =a + (n – 1)d, the nth partial sum is given by either of these formulas.
Substituting in Formula 2 for the partial sum of an arithmetic sequence, we get:
Ex 7. An amphitheater has 50 rows of seats with 30 seats in the first row, 32 in the second, 34 in the third, and so on. Find the total number of seats.
We are asked to find n when Sn = 572.