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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, AMC-12, SAT-II, Math Kangaroo 9-10.
'Geometric progression' is another term for 'geometric sequence'. In fact, it is the more traditional terminology.
If each term of a sequence is in a constant ratio with the previous (and with the next) we call the sequence geometric.
The factor by which we multiply a term to obtain the next term is called the common ratio.
We can build the entire sequence if we know the first term of the sequence and the common ratio.
The N-th term of the geometric sequence:
Let a1 be the first term of the sequence and r the common ratio.
|Rank of term|
|Name of term|
It is easy to see from the table that the N-th term would be:
The sum of the geometric sequence
We want to compute the sum of the first N terms:
By replacing all the terms with their value given by the N-th term formula, we get:
We see that the first term is a factor common to all the terms. We factor it out:
In the parenthesis we see an expression that depends only on r. This sum has N terms (from rank 0 to rank N-1). Let us denote this sum with TN(r):
Let us know multiply the entire sum by r:
and now subtract the two equalities:
where from we can solve for TN:
and multiply by the first term to go back to SN
Useful facts: Given a sequence of three consecutive terms of a geometric sequence, the middle term is the geometric mean of its neighbors.
Denote the middle term with m, then the three consecutive terms look like this:
Obviously, the middle term is the geometric mean:
Note: the geometric mean of two numbers is the square root of their product:
Extension to geometric series
A geometric series is a geometric sequence with an infinite number of terms.
This is an interesting mathematical notion because one can show that, in certain cases, the sum of such an infinity of terms can actually be a finite number. Of course, this is not true for any kind of arithmetic series, since regardless whether the common difference is positive or negative, the terms will eventually become very large in absolute value and the sum of such terms will become infinitely positive or infinitely negative.
When we can obtain finite results for the sum of a geometric series we say that the series converges. If, contrariwise, the sum goes to infinity in absolute value, we say that the series diverges.
To see when a series converges, look at the nature of the ratio r:
If |r|<1 (that is, if the ratio is a positive or negative number that is smaller than 1) then:
and the larger the rank of the term the closer the term is to zero.
Eventually, each term that is added on is always smaller and smaller and the sum ends up growing only by an 'infinitesimal' amount.
The point is that this sum will not be able to exceed a certain value, called a 'limit'.
We find this limit by making the rank N infinitely large in:
and the limit of the sum becomes:
Note that the sum never reaches the limit value, it only gets closer and closer to it either from values larger than the limit or from values smaller than the limit (depending on the sign of the ratio).