A1 1 R 4 N 5 Geometric Sequence Calculator

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This calculator finds the nth term of a geometric sequence given the first term, common ratio, and term number.

What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio (r). The sequence can be written as:

a₁, a₁r, a₁r², a₁r³, ..., a₁rⁿ⁻¹

Where:

a₁ is the first term
r is the common ratio
n is the term number

In this calculator, we're specifically looking at the sequence where a₁ = 1, r = 4, and n = 5.

Geometric sequence formula

The nth term of a geometric sequence can be calculated using the formula:

aₙ = a₁ × r^(n-1)

Where:

aₙ is the nth term
a₁ is the first term
r is the common ratio
n is the term number

For our specific case where a₁ = 1, r = 4, and n = 5:

a₅ = 1 × 4^(5-1) = 4⁴ = 256

Worked example

Let's calculate the 5th term of a geometric sequence where the first term is 1 and the common ratio is 4.

Identify the values:
- First term (a₁) = 1
- Common ratio (r) = 4
- Term number (n) = 5
Apply the formula:
a₅ = 1 × 4^(5-1) = 4⁴
Calculate the exponent:
4⁴ = 4 × 4 × 4 × 4 = 256
Final result: The 5th term is 256.

You can verify this by writing out the sequence:

1, 4, 16, 64, 256

FAQ

What is the difference between arithmetic and geometric sequences?: An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms.
Can the common ratio be negative?: Yes, the common ratio can be any real number except zero. A negative ratio creates an alternating sequence.
What happens if the common ratio is 1?: If the common ratio is 1, all terms in the sequence are equal to the first term.
How do I find the common ratio if I know two terms?: Divide the second term by the first term to find the common ratio.
What is the sum of a geometric sequence?: The sum of the first n terms of a geometric sequence can be calculated using the formula Sₙ = a₁(1 - rⁿ)/(1 - r) when r ≠ 1.