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(10) Arithmetic Sequences SUM(II).notebook
SERIES
Recall: A sequence is an ordered list of numbers.
The sum of the terms of a sequence is called a series. To find the sum of a certain number of terms of an arithmetic sequence:
The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
where Sn is the sum of n terms (nth partial sum),
a1 is the first term, an is the nth term.
To find the sum of arithmetic series, the formula for a­n may be used to find needed information .
1. Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...
To use the sum formula, an needs to be found first. a1 = 4, n = 20, d = 2 an = a1 + (n ­ 1)d
a20 = 4 + (20 – 1)(2)
Remember d is a20 = 4 + 19(2)
the common a­20 = 4 + 38
difference.
a20 = 42
Now the sum formula can be used.
n = 20, a1 = 4, an = a20 = 42
S20 = S20 = S20 = S20 = 460
(10) Arithmetic Sequences SUM(II).notebook
2. Determine the sum of the first 11 terms of the arithmetic sequence whose first 4 terms are 8, 11, 14, 17
a1 = 8, n = 11, d = 3, a11 = ? an = a1 + (n ­ 1)d
a11 = 8 + ( 11– 1)(3)
a11 = 8 + 10(3)
a­11 = 8 + 30
a11 = 38
n = 11, a1 = 8, a11 = 38, S11 = ?
n(a1 + an)
Sn = S11 = S11 = 2
11(8 + 38)
2
11(46)
2
S11 = 506
2
S11 = 253
3. Find the sum of the arithmetic series 3, 6, 9, .... ,99
To find the sum you need to know
the first and last terms and n.
Find n!!!!!
a1 = 3, an = 99, d = 3 n = ?
an = a1 + (n ­ 1)d
99 = 3 + (n ­ 1)(3)
99 = 3 + 3n ­ 3
99 = 3n 33 = n
Now, the sum!!!
a1 = 3, a33 = 99, n = 33, S33 = ?
Sn = S33 = n(a1 + an)
2
33(3 + 99)
2
33(102)
S33 = 2
3366
S33 = 2
S33 = 1683
(10) Arithmetic Sequences SUM(II).notebook
4. Determine the sum of 22, 16, 10, … , ­80
a1 =22, an = ­80, d = ­6 n = ?
a1 = 22, a18 = ­80, n = 18, S18 = ?
an = a1 + (n ­ 1)d
­80 = 22 + (n ­ 1)(­6)
­80 = 22 ­ 6n + 6
­80 = ­6n + 28 ­108 = ­6n
18 = n
n(a1 + an)
Sn = S18 = 2
18(22 + ­80)
2
18(­58)
S18 = 2
S18 = ­1044
2
S18 = ­522