Geometric Sequence and Sum: Exploring the Formula and Example Calculation

The Geometric Sequence Finite Sum

In mathematics, a geometric sequence is a sequence of numbers in which each term is found by multiplying the previous term by a constant factor called the common ratio (r)

In mathematics, a geometric sequence is a sequence of numbers in which each term is found by multiplying the previous term by a constant factor called the common ratio (r). The formula to find the nth term (Tn) of a geometric sequence is:

Tn = ar^(n-1)

where a is the first term and n is the term number.

To find the sum of a finite geometric sequence, we can use the formula for the sum of a geometric series:

S = (a(1 – r^n))/(1 – r)

where S is the sum of the sequence.

Let’s consider an example to illustrate this:

Example: Find the sum of the geometric sequence 2, 6, 18, 54, 162.

In this example, we are given a = 2 (the first term) and r = 3 (the common ratio). We can see that this sequence is growing by multiplying each term by 3.

To find the sum, we need to know the number of terms (n). In this case, we have 5 terms (2, 6, 18, 54, 162).

Using the formula for the sum of a geometric series, we can substitute the given values into the formula:

S = (2(1 – 3^5))/(1 – 3)
S = (2(1 – 243))/(-2)
S = (2(-242))/(-2)
S = 242

Therefore, the sum of the geometric sequence 2, 6, 18, 54, 162 is 242.

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