Geometric Series
How to find the value of a geometric series (sum of a geometric sequence): formula, 3 examples, and their solutions.
Formula
A series means
the sum of the terms of a sequence.
Sn = a1 + a2 + a3 + ... + an
For an geometric sequence,
Sn = [a(rn - 1)]/[r - 1].
Sn: The sum from a1 to an
a: First term, a1
r: Common ratio
Use this formula when |r| > 1.
(When r = 1, Sn = an.)
Sn = [a(1 - rn)]/[1 - r].
Use this formula when |r| < 1.
Examplea = 3, r = 2, n = 7, S7 = ?
a = 3
r = 2
n = 7
Then S7 = [3(27 - 1)]/[2 - 1].
27 = 128
Power
2 - 1 = 1
128 - 1 = 127
3⋅127 = 381
So 381 is the answer.
Examplea1 = 4, a4 = 36, S5 = ?
a1 = 4
So a = 4.
For a geometric sequence,
a3 = a⋅r2.
So
a3 = 4⋅r2.
a3 = 36
So
a3 = 4r2 = 36.
Solve 4r2 = 36.
Divide both sides by 4.
Then r2 = 9.
9 = 32
Square root both sides.
Then r = ±3.
See the two cases.
Case 1: r = 3
a = 4
r = 3
Then S5 = [4(35 - 1)]/[3 - 1].
35 = 243
3 - 1 = 2
Cancel the denominator 2
and reduce the 4 in the numerator to, 4/2, 2.
243 - 1 = 242
2⋅242 = 484
So S5 = 484
for case 1.
See the other case.
Case 2: r = -3
a = 4
r = -3
Then S5 = [4((-3)5 - 1)]/[-3 - 1].
35 = 243
So (-3)5 = -243.
-3 - 1 = -4
4/(-4) = -1
-243 - 1 = -244
-(-244) = 244
So S5 = 244
for case 2.
Case 1: r = 3
S5 = 484
Case 2: r = -3
S5 = 244
So S5 = 484 or 244.
So
484 or 244
is the answer.
Example∑n = 14 5⋅(2/3)n
See the sigma notation.
n goes from 1.
So the first term, a1, is
when n = 1.
So a1 = 5⋅(2/3)1.
5⋅(2/3)1
= 5⋅(2/3)
= 10/3
So a1 = 10/3.
See the term in the sigma:
5⋅(2/3)n.
As n increases,
(2/3) is multiplied:
5⋅(2/3)1, 5⋅(2/3)2, 5⋅(2/3)3, ... .
So the given summation is a geometric series.
So r = 2/3.
n goes from 1 to 4.
So there are 4 terms.
So n = 4.
(This n is for
finding the geometric series S4.)
a = a1 = 10/3
r = 2/3
n = 4
Then S4 = [(10/3)[1 - (2/3)4]]/[1 - 2/3].
|r| < 1
So it's good to use the second formula.
(2/3)4
= 24/34
= 16/81
Power of a Quotient
1 = 3/3
1 = 81/81
3/3 - 2/3 = 1/3
Multiply 3
to both of the numerator and the denominator.
[10/3]⋅3 = 10
[1/3]⋅3 = 1
81/81 - 16/81 = 65/81
10⋅[65/81] = 650/81
So 650/81 is the answer.