The Geometric Sequence Infinite Sum
The infinite sum of a geometric sequence refers to the sum of all terms in the sequence, which goes on indefinitely
The infinite sum of a geometric sequence refers to the sum of all terms in the sequence, which goes on indefinitely. A geometric sequence is a sequence in which each term is found by multiplying the previous term by a constant value called the common ratio (r).
The formula for finding the sum of an infinite geometric sequence, also known as the infinite sum, is as follows:
S = a / (1 – r)
Here, “S” represents the sum of the infinite geometric sequence, “a” represents the first term of the sequence, and “r” represents the common ratio.
It is important to note that for the sum to exist, the absolute value of the common ratio (|r|) must be less than 1.
To understand why this formula works, let’s consider a geometric sequence with a common ratio of “r” and a first term of “a”. The sequence would look like this:
a, ar, ar^2, ar^3, ar^4, …
To find the sum of this sequence, we can start by multiplying the entire sequence by the common ratio “r”:
r(a, ar, ar^2, ar^3, ar^4, …)
This gives us:
ar, ar^2, ar^3, ar^4, ar^5, …
Now, if we subtract the original sequence from this new sequence, term by term, we get:
r(a, ar, ar^2, ar^3, ar^4, …)
– (a, ar, ar^2, ar^3, ar^4, …)
= (0, ar^2 – ar, ar^3 – ar^2, ar^4 – ar^3, ar^5 – ar^4, …)
Notice that all the terms except the first one cancel out. We are left with:
0 + (ar^2 – ar) + (ar^3 – ar^2) + (ar^4 – ar^3) + (ar^5 – ar^4) + …
If we group the terms in parentheses, we get:
0 + (ar^2 – ar) + (ar^3 – ar^2) + (ar^4 – ar^3) + (ar^5 – ar^4) + …
= ar^2 + ar^3 + ar^4 + ar^5 + …
Now, notice that this new expression is almost identical to the original geometric sequence, except that it starts from the second term. If we let this sum be denoted as “S”, we can rewrite it as:
S = ar^2 + ar^3 + ar^4 + ar^5 + …
However, this new sequence is just a scaled version of the original sequence. We can obtain it by multiplying the original sequence by “r” and starting it from the second term. Thus, we have:
S = r(a, ar, ar^2, ar^3, ar^4, …)
= r(a + ar + ar^2 + ar^3 + ar^4 + …)
= ra + rar + rar^2 + rar^3 + rar^4 + …
Now, if we subtract this new expression from the original geometric sequence, term by term, we get:
(a, ar, ar^2, ar^3, ar^4, …)
– (ra + rar + rar^2 + rar^3 + rar^4 + …)
= (a – ra) + (ar – rar) + (ar^2 – rar^2) + (ar^3 – rar^3) + (ar^4 – rar^4) + …
Again, we see that all the terms except the first one cancel out. We are left with:
(a – ra) + (ar – rar) + (ar^2 – rar^2) + (ar^3 – rar^3) + (ar^4 – rar^4) + …
= a – ra
We know that the sum of the infinite sequence is just the original sequence minus the scaled version. Therefore:
S = a – ra
This equation can be rewritten as:
S = a(1 – r)
Finally, if we solve this equation for “S”, we get the formula for the sum of an infinite geometric sequence:
S = a / (1 – r)
It is essential to remember that this formula is only valid when the absolute value of the common ratio (|r|) is less than 1. If the absolute value of the common ratio is equal to or greater than 1, the sum of the infinite geometric sequence does not exist.
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