Note that a series is an indicated sum of the terms of a
sequence!! In this section, we work only with finite series and
the related sums.
How to find
the sum of a finite Arithmetic Series!
s_{n}= n(t_{1} + t_{n})/2
To
find the sum of a finite arithmetic series, you need to know three
things. The first term, the last term and the number of terms.
Example problem:
1)
Find the sum of the first 30 terms of 5 + 9 + 13 + 17 + ^{. . .}
Answer:
n = 30, that's pretty obvious!
t_{1}
= 5,
and that's pretty obvious!
We need the 30th term. Use the defintion of an arithmetic
sequence.
t_{30}
= 5 + 29^{.}4 = 121
Therefore:
S_{30} = 30(5 +
121)/2 = 1890
How to find
the sum of a finite Geometric Series
S_{n} =
t_{1}(1 - r^{n})/(1 - r) where r is the common ratio
and
(r doesn't = 0)
To
find the sum of a finite geometric series, you need to know three
things: the first term, how many terms to add and the common
ratio!! (piece of cake!)
Example problem:
2)
Find the sum of the first 10 terms of the geometric series: 4, 8, 16, 32, 64, ^{.
.
.}
Answer:
t_{1} = 4
r = 2
t_{10}
= 4 ^{.} 2^{9} = 2048 (This is the formula for
a
geometric sequence!)
On to the next section! We will begin out study of
limits in the next section as related to infinite sequences. I
think we must be getting close to some calculus. What do you
think? See you in the next section!
Current quizaroo #
13a
1) Find the formula
for t_{n} only if it is an arithmetic
sequence: 3, 7, 11, 15, 19,
. . .
a)
not arithmetic
b)
t_{n} = 4n - 1
c) t_{n} = 3n + 1
d) t_{n} = t_{n-1} + n
e) t_{n} = 3n - 1
2) Give the formula
for t_{n} only if it is a geometric
sequence: 2, 5, 10, 17, 26, 37,
. . .
a) not geometric
b)
t_{n} = n^{2} + 1
c)
t_{n} = 2n + 1
d)
t_{n} = t_{n-1} + 2n + 3
e)
t_{n} = n^{2} - 1
3) Find the recursive formula for the sequence:
3, 13, 33, 73, 153, . . .
a)
t_{1} = 3, t_{n} = t_{n-1} + 10n
b)
t_{n} = 2t_{n-1} + 7
c)
t_{1} = 3, t_{n} = 2t_{n-1} + 7
d)
t_{n} = 3 + 10n
e)
There is no formula
4) Find the sum
of the
arithmetic series: S_{50}; 6
+
12 + 18 + 24 + 30 + ^{. . . }
a) 15300
b) 306
c) 153
d) 7500
e) 7650
5) Find the sum
of all the multiples of 4 between 1 and
999.