GEOMETRIC SEQUENCES & SERIES

Geometric Drill

The four moves: find r & a₁ · sigma notation · finite sums · infinite sums. Learn the process, then use the generator to drill until it's automatic.

‹ Back to all modules

Reference

The 4 Formulas

Geometric sequences MULTIPLY by the same ratio $r$ each step. Everything else follows from that.

nth Term

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

Terms multiply by $r$ each step. The exponent is one less than the term number.

Finite Sum (first n terms)

$S_n = \dfrac{a_1(1 - r^n)}{1 - r}$

Works for any $r \neq 1$. Order matters: $(1-r^n)$ over $(1-r)$, not reversed.

Infinite Sum

$S = \dfrac{a_1}{1 - r}$

Only valid when $|r| < 1$ — when terms shrink toward zero.

Sigma Notation

$\displaystyle\sum_{k=s}^{e} (\text{formula})$

$\Sigma$ = "add up." Plug in $k = s, s+1, \ldots, e$ and sum all the results.

Worked Example 1

Find r and a₁ from Two Terms

Given: The 4th term is 24 and the 7th term is 192. Find $r$ and $a_1$.

Step 1 — Count the jumps. From term 4 to term 7 = 3 jumps (7 − 4 = 3). Each jump multiplies by $r$, so $a_7 = a_4 \cdot r^3$.

Step 2 — Set up the equation. $192 = 24 \cdot r^3$

Step 3 — Solve for $r^3$. $r^3 = \dfrac{192}{24} = 8$

Step 4 — Cube root. $r = \sqrt[3]{8} = 2$

Step 5 — Back-substitute to find $a_1$. Use $a_4 = a_1 \cdot r^3$: $\;24 = a_1 \cdot 8 \;\Rightarrow\; a_1 = 3$

Exam warning: On your exam the numbers will be uglier — fractions, negatives. The process is identical. It's the fraction arithmetic that grinds you, not the concept. Practice the process here until it's automatic, then the ugly numbers are just arithmetic.

Worked Example 2

Sigma Notation — Expand and Evaluate

Given: Evaluate $\displaystyle\sum_{k=0}^{5} 3 \cdot 2^k$

Step 1 — What does $\Sigma$ mean? Plug in each value of $k$ from 0 to 5 and add them all up.

Step 2 — Expand. $3 \cdot 2^0 + 3 \cdot 2^1 + 3 \cdot 2^2 + 3 \cdot 2^3 + 3 \cdot 2^4 + 3 \cdot 2^5$

Step 3 — Compute each power. $= 3(1) + 3(2) + 3(4) + 3(8) + 3(16) + 3(32)$

Step 4 — Factor and add. $= 3(1 + 2 + 4 + 8 + 16 + 32) = 3(63) = 189$

Shortcut check: This is a geometric sum with $a_1 = 3$, $r = 2$, $n = 6$ terms. $S_6 = \dfrac{3(1 - 2^6)}{1 - 2} = \dfrac{3(-63)}{-1} = 189$ ✓

Your specific error to watch: $2^5 \neq 25$. That's $5^2$. Powers of 2 just keep doubling: 1, 2, 4, 8, 16, 32, 64, 128. If you ever get a power-of-2 answer that looks like a two-digit number with a 5 in it, stop and recount.

Worked Example 3

Finite Geometric Sum

Given: Find $S_{10}$ for the geometric series with $a_1 = 4$ and $r = \frac{1}{2}$.

Step 1 — Identify. $a_1 = 4$, $r = \tfrac{1}{2}$, $n = 10$. Use $S_n = \dfrac{a_1(1 - r^n)}{1 - r}$.

Step 2 — Compute $r^n$. $\left(\tfrac{1}{2}\right)^{10} = \dfrac{1}{1024}$. When $r$ is a fraction, $r^n$ gets tiny. Don't panic.

Step 3 — Plug in. $S_{10} = \dfrac{4\!\left(1 - \tfrac{1}{1024}\right)}{1 - \tfrac{1}{2}} = \dfrac{4 \cdot \tfrac{1023}{1024}}{\tfrac{1}{2}}$

Step 4 — Dividing by $\frac{1}{2}$ = multiplying by 2. $= 4 \cdot \dfrac{1023}{1024} \cdot 2 = \dfrac{8 \cdot 1023}{1024} = \dfrac{8184}{1024} \approx 7.992$

Two traps here: (1) The denominator is $1 - r$, not $r - 1$ — order matters for the sign. If you flip it, your answer is negated. (2) Keep-change-flip when dividing by a fraction: $\div \frac{1}{2}$ means $\times 2$.

Worked Example 4

Infinite Geometric Series

Given: Find the sum $\displaystyle\sum_{k=1}^{\infty} 6 \cdot \left(\tfrac{1}{3}\right)^{k-1}$

Step 1 — Identify $a_1$ and $r$. When $k=1$: $6 \cdot (1/3)^0 = 6$, so $a_1 = 6$. The ratio $r = \tfrac{1}{3}$.

Step 2 — Check $|r| < 1$. $\left|\tfrac{1}{3}\right| = \tfrac{1}{3} < 1$ ✓ — the infinite sum formula is valid.

Step 3 — Apply the formula. $S = \dfrac{a_1}{1 - r} = \dfrac{6}{1 - \tfrac{1}{3}} = \dfrac{6}{\tfrac{2}{3}}$

Step 4 — Keep-change-flip. $\dfrac{6}{\tfrac{2}{3}} = 6 \times \dfrac{3}{2} = 9$

If $|r| \geq 1$, stop. The infinite sum is undefined — the terms don't shrink to zero, they grow (or stay the same). You'll get a nonsense answer if you apply the formula anyway. Always check $|r| < 1$ first.

Practice Generator

Generate a Problem

Pick a type, work it on paper, then reveal the solution. Aim for 3 in a row without peeking.

Session: — / —

Hit "New Problem" to start.

Aaron's Trap Sheet

Errors from your last session. Read these before every practice set.

⚠ $2^5 \neq 25$. That's $5^2$. Powers of 2 just double: 2, 4, 8, 16, 32, 64, 128. In doubt? Start at 1 and double each time.

⚠ Negative base, even exponent = POSITIVE. $(-2)^2 = 4$, not $-4$. $(-2)^4 = 16$. Even exponent always kills the negative.

⚠ Jumps between terms: subtract the indices. From term 5 to term 8 = $8 - 5 = \mathbf{3}$ jumps (not 4). You're counting multiplications, not terms.

⚠ Cube roots to memorize. $\sqrt[3]{8} = 2$, $\sqrt[3]{27} = 3$, $\sqrt[3]{64} = 4$, $\sqrt[3]{125} = 5$, $\sqrt[3]{216} = 6$. These show up constantly.

⚠ Dividing by a fraction: keep-change-flip. $a \div \dfrac{b}{c} = a \times \dfrac{c}{b}$. Whenever you divide by $\frac{2}{3}$, you're really multiplying by $\frac{3}{2}$.

‹ Back to all modules