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Arithmetico-Geometric sequence

As the name suggests, an arithmetico-geometric sequence is formed by the combination of an arithmetic and a geometric progression.

What is an arithmetico-goemtric sequence?
An arithmetico-geometric sequence is a sequence of numbers, each number of which is formed by multiplying the corresponding numbers of an arithmetic progression and a geometric progression.

For example,
• an arithmetic progression is 2, 5, 8, 11, ...
• and a geometric progression is 2, 4, 8, 16 ...

On multiplying each corresponding number of the above arithmetic and geometric progressions, we get the following numbers:
4, 20, 64, 176, ...
In the above arithmetico-geometric sequence,
• the first number 4 is obtained by multiplying the first numbers of the arithmetic and geometric progressions above: 2 x 2 = 4
• the second number 20 is obtained by multiplying the first numbers of the arithmetic and geometric progressions above: 5 x 4 = 20
• and so on...

General form or standard form of an arithmetico-geometric sequence
Let an arithmetic progression be represented as follows:
a, a + d, a + 2d, a + 3d, ...
and let a geometric progression be represented as follows:
1, r, r2, r3, ...
Then, the corresponding arithmetico-geometric sequence is formed by multiplying each corresponding terms of the above two progressions as follows:
 General form of arithmetico-geometric sequence: a, (a + d)r, (a + 2d)r2, (a + 3d)r3, ...
This is the general form (or standard form) of an arithmetico-geometric sequence.

General term of an arithmetico geometric sequence

As discussed above, each term of an arithmetico-geometric sequence is formed by multiplying the corresponding terms of an arithmetic and a geometric sequence.

Thus, since the general term of an arithmetic progression is
Tn = a + (n - 1)d
and the general term of a geometric progression is
Tn = r(n - 1)
therefore the general term of an arithmetico geometric sequence is

 General term of arithmetico-geometric sequence: Tn = (a + (n - 1)d) * r(n - 1)
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1. Wow.. Very well explained. Thank You a lot. Looking at your other posts, You explain such topics superbly. Thank You again..

2. 2,4,6,8 is not geometric progression

1. Corrected it... meant it to be 2, 4, 8, 16. Thank you for pointing that out.

3. exceptionally good!

4. Nice..... i like the way of explaination

5. Thanks for the explanation, friendly blogger!

a) Please consider deleting the "1" and writing an "r" instead. (The parasitic "1" appears after the sentence "and let a geometric progression be represented as follows").

b) Please consider inserting the missing r's in both the image and description of the General Form.

Have a nice day!

1. Thanks for your comment! I would like to point out that the '1' is the first term of the geometric progression. It gets multiplied with the first term of the arithmetic progression (which is 'a') so we get a * 1 = a as the first term of the arithemetico-geometric progression.

However, I missed writing the first term of the arithemetico-geometric progression in the image, which I have now corrected above.