A geometric sequence within a arithmetic sequence

[B4ssHunter]

the main question here is that can a sequence * arithmetic * be correct if the difference is also changing in terms of a geometric sequence ?\
now look at this sequence
0.33,0.3333,0.333333
now if we calculate the difference between the first two terms
its 0.0033
between the second and the third its
0.000033
the difference between the numbers goes as a geometric sequence
0.33,0.0033,0.000033 and the Ratio between them is 1/100
can this work as a sequence ? and if so what kind of sequence is it ?

 

[Stephen Tashi]

the main question here is that can a sequence * arithmetic * be correct if the difference is also changing in terms of a geometric sequence ?

What do you mean by "correct"?

 

[junaid314159]

There are many different types of sequences. Some of the most common ones that we study are arithmetic and geometric sequences. Arithmetic sequences have a common difference while geometric sequences have a common ratio. If a sequence does not have a common difference or a common ratio then it is neither arithmetic or geometric but it is still a sequence.

For example:
1,2,3,4,5,6,... is arithmetic but not geometric
1,2,4,8,16,... is geometric but not arithmetic
1,4,9,16,25,... is neither arithmetic or geometric but is still a sequence

The sequence you mentioned is not arithmetic because it does not have a common difference. It is not geometric because there is no common ratio between terms. Actually, what you have is a sequence of partial sums of a geometric series.

For example:
S1 = 0.33
S2 = 0.33 + 0.0033
S3 = 0.33 + 0.0033 + 0.000033
etc.
So each partial sum individually is a geometric series but the sequence of partial sums is not arithmetic or geometric.

Hope that makes sense,
Junaid Mansuri

 

[B4ssHunter]

There are many different types of sequences. Some of the most common ones that we study are arithmetic and geometric sequences. Arithmetic sequences have a common difference while geometric sequences have a common ratio. If a sequence does not have a common difference or a common ratio then it is neither arithmetic or geometric but it is still a sequence.

For example:
1,2,3,4,5,6,... is arithmetic but not geometric
1,2,4,8,16,... is geometric but not arithmetic
1,4,9,16,25,... is neither arithmetic or geometric but is still a sequence

The sequence you mentioned is not arithmetic because it does not have a common difference. It is not geometric because there is no common ratio between terms. Actually, what you have is a sequence of partial sums of a geometric series.

For example:
S1 = 0.33
S2 = 0.33 + 0.0033
S3 = 0.33 + 0.0033 + 0.000033
etc.
So each partial sum individually is a geometric series but the sequence of partial sums is not arithmetic or geometric.

Hope that makes sense,
Junaid Mansuri

right that makes sense , i was thinking of this because my teacher told me once about how to make a sequence of a 3.3333333and so on , i didn't think of it as getting the sum at the end but rather each term would be 0.33 ,0.3333 and so on
so 0.333333333~ is actually a geometric sequence
where it goes like 0.3,0.03,0.003 and then in the end if we take the sum of the series it gives 0.3333333

 

[CellCoree]

Yes, this can work as a sequence and it is a type of mixed sequence, specifically a combination of an arithmetic sequence and a geometric sequence. In an arithmetic sequence, the difference between consecutive terms remains constant, while in a geometric sequence, the ratio between consecutive terms remains constant. In this case, the difference between terms is changing in a geometric pattern, while still following an overall arithmetic pattern. This type of sequence is commonly referred to as a "mixed arithmetic-geometric sequence" or a "hybrid sequence." It is a valid sequence and can be used in various mathematical and scientific calculations.