How Do You Find the Nth Term and Sum of a Geometric Sequence?

Use the first term and common ratio. For a geometric sequence with first term $u_1$ and common ratio $r$, the nth term is $u_n=u_1r^{n-1}$, while the finite sum is $S_n=\frac{u_1(1-r^n)}{1-r}$ when $r\neq 1$.

The Reasoning

A sequence is geometric when every term after the first is obtained by multiplying the previous term by the same common ratio. Find the ratio using two consecutive terms:

$$r=\frac{u_{k+1}}{u_k}.$$

The nth-term formula follows the pattern

$$u_1,\quad u_1r,\quad u_1r^2,\quad u_1r^3,\ldots$$

The exponent is $n-1$ because moving from the first term to the nth term requires exactly $n-1$ multiplications by $r$. The sum formula combines all terms from $u_1$ through , rather than selecting only one term.

To see why the sum formula works, write $S_n=u_1+u_1r+\cdots+u_1r^{n-1}$ and multiply every term by : . Subtracting the second equation from the first cancels the middle terms: . Dividing by gives the finite-sum formula. This derivation also explains why must be treated separately: every term then equals , so the sum is .

Worked Example

Consider $3,6,12,24,\ldots$. Here,

$$u_1=3,\qquad r=\frac{6}{3}=2.$$

The sixth term is

$$u_6=3(2)^{6-1}=3(32)=96.$$

The sum of the first six terms is

$$S_6=\frac{3(1-2^6)}{1-2}=189.$$

Therefore, $u_6=96$, but $S_6=189$. This distinction matters: $u_6$ is one term, whereas includes .

A quick check is useful: because all six terms are positive, the sum must exceed the sixth term. Also, directly adding the terms gives $189$, confirming the formula and helping detect a GDC entry error.

Exam Technique

This is SL 1.3, Geometric Sequences and Series, common to AI SL and HL, and it may appear on either GDC-active paper. State $u_1$, $r$, and $n$, then write the appropriate formula before evaluating it with your GDC.

A common misconception is using $r^n$ for the nth term. Use $r^{n-1}$; reserve $r^n$ for the finite-sum formula. Keep full GDC precision and round only the final answer when instructed.