Geometric sequence sum formula
What is the Geometric Sequence Sum Formula?
Answer:
A geometric sequence (or geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of the first ( n ) terms of a geometric sequence can be calculated using the geometric sequence sum formula. Here is a detailed explanation of the formula and how it is derived.
1. Geometric Sequence Definition
A geometric sequence can be represented as follows:
a, ar, ar^2, ar^3, \ldots, ar^{n-1}
where:
- ( a ) is the first term,
- ( r ) is the common ratio,
- ( n ) is the number of terms.
2. Geometric Sequence Sum Formula
The sum ( S_n ) of the first ( n ) terms of a geometric sequence can be given by the formula:
S_n = a \frac{1 - r^n}{1 - r} \quad \text{for} \, r \neq 1
3. Derivation of the Formula
To understand how this formula is derived, follow these steps:
-
Write the sum of the first ( n ) terms:
S_n = a + ar + ar^2 + ar^3 + \cdots + ar^{n-1} -
Multiply both sides of the above equation by the common ratio ( r ):
rS_n = ar + ar^2 + ar^3 + \cdots + ar^{n-1} + ar^n -
Subtract the second equation from the first equation:
S_n - rS_n = a + ar + ar^2 + ar^3 + \cdots + ar^{n-1} - (ar + ar^2 + ar^3 + \cdots + ar^{n-1} + ar^n)Simplifying, we have:
S_n(1 - r) = a - ar^n -
Isolate ( S_n ):
S_n = a \frac{1 - r^n}{1 - r}
4. Important Special Case: ( r = 1 )
When ( r = 1 ), every term in the sequence is equal to ( a ). Thus, the sum of the first ( n ) terms is simply:
S_n = na
Application
Suppose we have a geometric sequence with the first term ( a = 3 ) and a common ratio ( r = 2 ), and we want to find the sum of the first 5 terms.
-
Determine the values:
- ( a = 3 )
- ( r = 2 )
- ( n = 5 )
-
Apply the formula:
S_5 = 3 \frac{1 - 2^5}{1 - 2} = 3 \frac{1 - 32}{-1} = 3 \times 31 = 93
Final Answer:
The sum of the first 5 terms of the geometric sequence is ( 93 ).
By deriving and understanding the formula in this way, one can efficiently find the sum of the first ( n ) terms of any geometric sequence, provided the necessary parameters are known.