Finding the nth Term of a Geometric Sequence

What is a geometric sequence?

A geometric sequence has a constant ratio $r$ between consecutive terms. Each term is obtained by multiplying the previous term by $r$:

$a_1, \quad a_1 r, \quad a_1 r^2, \quad a_1 r^3, \quad \ldots$

The formula

$a_n = a_1 \cdot r^{n-1}$

Step-by-step solution: Find the 8th term of 2, 6, 18, 54, ...

Step 1 — Identify $a_1$ and $r$.

$a_1 = 2$. Common ratio: $r = 6/2 = 3$ (each term is 3× the previous).

Step 2 — Apply the formula with $n = 8$:

$a_8 = 2 \cdot 3^{8-1} = 2 \cdot 3^7$

Step 3 — Compute $3^7$: $3^1 = 3$, $3^2 = 9$, $3^3 = 27$, $3^4 = 81$, $3^5 = 243$, $3^6 = 729$, $3^7 = 2187$.

Step 4 — Final answer: $a_8 = 2 \times 2187 = 4374$.

Check: The sequence is $2, 6, 18, 54, 162, 486, 1458, 4374$. Each term is 3× the previous ✓.

Arithmetic vs geometric

• Arithmetic: add a constant ($+d$). Growth is linear.
• Geometric: multiply by a constant ($\times r$). Growth is exponential.

Geometric sequences grow (or decay) much faster than arithmetic ones. After 20 terms with $r = 3$, the term exceeds 6 billion!

Common mistakes

• Using $r^n$ instead of $r^{n-1}$. The exponent is $(n-1)$, not $n$, because the first term $a_1$ already exists without any multiplication.
• Confusing $r$ with $d$. In geometric sequences you multiply; in arithmetic you add.

Try it in the visualization

Adjust $a_1$, $r$, and $n$. The bar chart shows exponential growth (or decay if $|r| < 1$). The ×$r$ label between bars shows the constant ratio.