Writing the Explicit Formula for a Geometric Sequence
The explicit formula for a geometric sequence aₙ = a₁ · rⁿ⁻¹ lets you find any term directly using the first term a₁ and the common ratio r — a key skill in enVision Algebra 1 Chapter 6 for Grade 11. For a sequence with a₁ = 3 and r = 2: aₙ = 3 · 2ⁿ⁻¹. For 5, 15, 45, ...: a₁ = 5, r = 15/5 = 3, so aₙ = 5 · 3ⁿ⁻¹. To verify: a₁ = 5·3⁰ = 5 ✓, a₂ = 5·3¹ = 15 ✓. The exponent (n-1) rather than n ensures that substituting n = 1 gives back the first term a₁, not a₁ multiplied by r.
Key Concepts
Property The explicit formula for a geometric sequence is used to find any term in the sequence directly. The formula is given by: $$a n = a 1 \cdot r^{n 1}$$ where $a n$ is the $n$ th term, $a 1$ is the first term, and $r$ is the common ratio.
Examples To write the explicit formula for a sequence with a first term of $3$ and a common ratio of $2$, we substitute $a 1 = 3$ and $r = 2$ into the formula: $a n = 3 \cdot 2^{n 1}$. For the sequence $5, 15, 45, \dots$, the first term is $a 1 = 5$. The common ratio is $r = \frac{15}{5} = 3$. The explicit formula is $a n = 5 \cdot 3^{n 1}$. For the sequence $100, 50, 25, \dots$, the first term is $a 1 = 100$ and the common ratio is $r = \frac{50}{100} = \frac{1}{2}$. The explicit formula is $a n = 100 \cdot (\frac{1}{2})^{n 1}$.
Explanation The explicit formula, also known as the general term, allows you to calculate any term in a geometric sequence without having to find all the preceding terms. It is defined using the sequence''s first term ($a 1$) and its common ratio ($r$). To write the formula, you simply substitute the known values of $a 1$ and $r$ into the standard equation $a n = a 1 \cdot r^{n 1}$. This formula is a type of exponential function, where the term number $n$ acts as the variable in the exponent.
Common Questions
What is the explicit formula for a geometric sequence?
aₙ = a₁ · rⁿ⁻¹, where a₁ is the first term and r is the common ratio. This formula finds the nth term directly without computing all preceding terms.
Write the explicit formula for a sequence with a₁ = 3 and r = 2.
aₙ = 3 · 2ⁿ⁻¹. Check: a₁ = 3·2⁰ = 3 ✓, a₂ = 3·2¹ = 6 ✓, a₃ = 3·2² = 12 ✓.
What is the explicit formula for 5, 15, 45, ...?
a₁ = 5 and r = 15/5 = 3. Formula: aₙ = 5 · 3ⁿ⁻¹.
How do you find the common ratio from a geometric sequence?
Divide any term by the previous term: r = a₂/a₁ = a₃/a₂ = ... For 2, 6, 18, 54: r = 6/2 = 3.
Why does the explicit formula use (n-1) as the exponent, not n?
At n=1, we want aₙ = a₁. Using (n-1): a₁ = a₁ · r⁰ = a₁ · 1 = a₁ ✓. If we used n, then a₁ = a₁ · r¹ = a₁r, which is wrong unless r=1.