Difference between revisions of "Arithmetico-geometric series"
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<math>x_n=(a_1+d(n-1))(g_1\cdot r^{n-1})</math> | <math>x_n=(a_1+d(n-1))(g_1\cdot r^{n-1})</math> | ||
− | Let <math>S_n</math> represent the sum of the first n terms. | + | Let <math>S_n</math> represent the sum of the first <math>n</math> terms. |
<math>S_n=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots+(a_1+(n-1)d)(g_1r^{n-1})</math> | <math>S_n=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots+(a_1+(n-1)d)(g_1r^{n-1})</math> | ||
Latest revision as of 19:39, 17 August 2020
An arithmetico-geometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined as: , where
and
are the
th terms of arithmetic and geometric sequences, respectively.
Finite Sum
The sum of the first terms of an
is
, where
is the common difference of
and
is the common ratio of
. Or,
, where
is the sum of the first
terms of
.
Proof:
Let represent the sum of the first
terms.
Infinite Sum
The sum of an infinite arithmetico-geometric sequence is , where
is the common difference of
and
is the common ratio of
(
). Or,
, where
is the infinite sum of the
.