Arithmetic and Geometric Series

➕ Arithmetic & Geometric Series

Part 1 of 5 — From Sequences to Series & Sigma Notation

Topics in This Part

🔑 Key Concept: A sequence is an ordered list of numbers. A series is what you get when you add the terms of a sequence. This lesson is about adding them up — quickly, with formulas, instead of one term at a time.

Sequence vs. Series

Start with the sequence $3,\ 7,\ 11,\ 15,\ 19$.

• As a sequence, we just list the terms.
• As a series, we add them: $3 + 7 + 11 + 15 + 19 = 55$.

A partial sum is the sum of the first $n$ terms, written $S_n$:

$S_1 = 3,\quad S_2 = 3+7 = 10,\quad S_3 = 3+7+11 = 21$

💡 The whole point of this lesson is to find $S_n$ without adding every term by hand. Adding $100$ numbers is slow; a formula is instant.

Concept Check 🎯

Sigma (Summation) Notation

Mathematicians abbreviate a sum with the Greek capital letter sigma, $\Sigma$:

$\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n$

Read it as "the sum of $a_k$ as $k$ goes from $1$ to $n$."

• $k$ is the index (a counter).
• The number below $\Sigma$ is the starting value of $k$.
• The number above $\Sigma$ is the ending value of $k$.
• $a_k$ is the rule — plug in each $k$ and add the results.

Example

$\sum_{k=1}^{4} (2k + 1) = \underbrace{3}_{k=1} + \underbrace{5}_{k=2} + \underbrace{7}_{k=3} + \underbrace{9}_{k=4} = 24$

⚠️ Watch the index range. $\sum_{k=1}^{4}$ has four terms ($k = 1, 2, 3, 4$). A sum like $\sum_{k=0}^{4}$ would have five terms. Count carefully: (top) − (bottom) + 1.

Decode the Notation 🔽

Consider the sum $\displaystyle\sum_{k=1}^{5} (3k - 2)$.

Evaluate the Sum 🧮

Write out the terms and add them.

1) $\displaystyle\sum_{k=1}^{3} (4k) = \,?$ 2) $\displaystyle\sum_{k=1}^{4} (k^2) = \,?$ 3) $\displaystyle\sum_{k=2}^{4} (k + 5) = \,?$

➕ Arithmetic & Geometric Series

Part 2 of 5 — The Arithmetic Series Formula

🔑 The Idea: An arithmetic series adds terms that change by a constant common difference $d$. There's a beautiful shortcut — pair the first and last terms — that sums them all at once.

Gauss's Pairing Trick

Legend says young Gauss added $1 + 2 + \cdots + 100$ in seconds. The trick: pair the ends.

Add the two lines term by term — every pair sums to $101$, and there are $100$ pairs:

$2S = 100 \times 101 = 10100 \;\Rightarrow\; S = \frac{10100}{2} = 5050$

This works for any arithmetic series, giving the formula:

$S_n = \frac{n}{2}\,(a_1 + a_n)$

💡 In words: the sum equals the number of terms times the average of the first and last term. ($a_1$ is the first term, $a_n$ is the last term, $n$ is how many terms.)

Concept Check 🎯

Worked Examples

Example 1 — first and last term known

Find $4 + 7 + 10 + \cdots + 31$ (common difference $d = 3$).

First find $n$. Since $a_n = a_1 + (n-1)d$: $31 = 4 + (n-1)(3) \;\Rightarrow\; 27 = 3(n-1) \;\Rightarrow\; n - 1 = 9 \;\Rightarrow\; n = 10$

Now apply the sum formula: $S_{10} = \frac{10}{2}(4 + 31) = 5 \times 35 = 175$

Example 2 — last term unknown

Sum the first $20$ terms of $5, 8, 11, 14, \dots$ ($a_1 = 5$, $d = 3$).

Find the $20$th term first: $a_{20} = 5 + (20-1)(3) = 5 + 57 = 62$. $S_{20} = \frac{20}{2}(5 + 62) = 10 \times 67 = 670$

⚠️ You need the last term $a_n$ to use $\frac{n}{2}(a_1 + a_n)$. If it's not given, compute it with $a_n = a_1 + (n-1)d$ first.

Build the Sum 🔽

You're summing the first $12$ terms of $2,\ 6,\ 10,\ 14,\dots$ ($a_1 = 2$, $d = 4$).

Sum the Series 🧮

1) $3 + 6 + 9 + \cdots + 30$ (that's $10$ terms). $S = \,?$ 2) First $25$ terms of $1, 3, 5, 7, \dots$ (odd numbers). $S = \,?$ 3) $\displaystyle\sum_{k=1}^{8}(2k+1)$, which is $3+5+\cdots+17$. $S = \,?$

➕ Arithmetic & Geometric Series

Part 3 of 5 — Finite Geometric Series

🔑 The Idea: A geometric series multiplies by a constant common ratio $r$ each step ($2 + 6 + 18 + 54 + \cdots$). Its sum also has a clean closed form.

The Finite Geometric Sum Formula

For a geometric series with first term $a_1$ and common ratio $r \ne 1$, the sum of the first $n$ terms is:

$S_n = a_1 \cdot \frac{1 - r^{\,n}}{1 - r}$

Where it comes from (the shift-and-subtract trick):

Subtract the second line from the first — almost everything cancels:

$S_n - r\,S_n = a_1 - a_1 r^{\,n} \;\Rightarrow\; S_n(1 - r) = a_1(1 - r^{\,n})$

Divide by $(1 - r)$ to get the formula.

💡 Identify $a_1$ (first term) and $r$ (divide any term by the one before it). To find $r$: $r = \dfrac{a_2}{a_1}$.

Concept Check 🎯

Worked Examples

Example 1 — $2 + 6 + 18 + 54 + 162$

Here $a_1 = 2$, $r = 3$, and there are $n = 5$ terms:

$S_5 = 2 \cdot \frac{1 - 3^5}{1 - 3} = 2 \cdot \frac{1 - 243}{-2} = 2 \cdot \frac{-242}{-2} = 2 \cdot 121 = 242$

✅ Check: $2 + 6 + 18 + 54 + 162 = 242$ ✓

Example 2 — a fractional ratio

Sum the first $4$ terms of $8 + 4 + 2 + 1 + \cdots$ ($a_1 = 8$, $r = \frac{1}{2}$):

$S_4 = 8 \cdot \frac{1 - \left(\frac{1}{2}\right)^4}{1 - \frac{1}{2}} = 8 \cdot \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 8 \cdot \frac{\frac{15}{16}}{\frac{1}{2}} = 8 \cdot \frac{15}{8} = 15$

✅ Check: $8 + 4 + 2 + 1 = 15$ ✓

Set Up the Geometric Sum 🔽

You're summing the first $5$ terms of $3 + 6 + 12 + 24 + \cdots$

Compute the Sum 🧮

1) $1 + 2 + 4 + 8 + 16 + 32$ ($a_1=1$, $r=2$, $n=6$). $S = \,?$ 2) $\displaystyle\sum_{k=1}^{3} 5\cdot 3^{\,k-1}$, i.e. $5 + 15 + 45$. $S = \,?$ 3) First $3$ terms of $9 + 3 + 1 + \cdots$ ($r = \frac{1}{3}$): $9 + 3 + 1$. $S = \,?$

➕ Arithmetic & Geometric Series

Part 4 of 5 — Infinite Geometric Series

🔑 The Big Surprise: You can add infinitely many numbers and still get a finite answer — but only for a geometric series when the terms shrink ($|r| < 1$).

When Does an Infinite Sum Have a Value?

Take $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \cdots$. The partial sums creep toward $1$:

$S_1 = 0.5,\quad S_2 = 0.75,\quad S_3 = 0.875,\quad S_4 = 0.9375,\ \dots \to 1$

They converge to a fixed number. This happens exactly when the common ratio satisfies $|r| < 1$ — each new term is a fraction of the one before, so the leftover shrinks to nothing.

In the formula $S_n = a_1\dfrac{1 - r^{\,n}}{1 - r}$, when $|r| < 1$ the term $r^{\,n} \to 0$ as $n \to \infty$. That leaves:

$S_\infty = \frac{a_1}{1 - r}\qquad (\text{valid only when } |r| < 1)$

⚠️ If $|r| \ge 1$ the series diverges — there is no finite sum, and you must not plug into $\frac{a_1}{1-r}$.

Concept Check 🎯

Worked Examples

Example 1 — $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots$

$a_1 = \frac{1}{2}$, $r = \frac{1}{2}$, and $|r| < 1$ so it converges:

$S_\infty = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$

Example 2 — a negative ratio

Sum $8 - 4 + 2 - 1 + \cdots$ Here $a_1 = 8$ and $r = -\frac{1}{2}$ (since $\frac{-4}{8} = -\frac{1}{2}$), and $|r| = \frac{1}{2} < 1$:

$S_\infty = \frac{8}{1 - \left(-\frac{1}{2}\right)} = \frac{8}{\frac{3}{2}} = 8 \cdot \frac{2}{3} = \frac{16}{3}$

Example 3 — repeating decimal as a series

$0.\overline{3} = 0.3 + 0.03 + 0.003 + \cdots$ is geometric with $a_1 = 0.3$, $r = 0.1$:

$S_\infty = \frac{0.3}{1 - 0.1} = \frac{0.3}{0.9} = \frac{1}{3}$

💡 That's why $0.\overline{3} = \frac{1}{3}$ — an infinite geometric series in disguise!

Converge or Diverge? 🔽

For each series, decide whether the infinite sum exists.

Find the Infinite Sum 🧮

Use $S_\infty = \dfrac{a_1}{1 - r}$ (fractions are fine, e.g. 16/3).

1) $a_1 = 4$, $r = \frac{1}{2}$: $S_\infty = \,?$ 2) $a_1 = 10$, $r = \frac{1}{5}$: $S_\infty = \,?$ 3) $9 + 3 + 1 + \frac{1}{3} + \cdots$ ($a_1 = 9$, $r = \frac{1}{3}$): $S_\infty = \,?$

➕ Arithmetic & Geometric Series

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read sigma notation, (2) sum an arithmetic series, (3) sum a finite geometric series, and (4) sum an infinite geometric series when it converges. Time to choose the right tool for each problem.

Quick Reference

Which type is it?

🔑 Arithmetic = constant difference (you add the same amount). Geometric = constant ratio (you multiply by the same amount). Check: is $a_2 - a_1$ constant, or is $a_2 / a_1$ constant?

⚠️ An infinite sum only has a value for a geometric series with $|r| < 1$. An infinite arithmetic series always diverges.

Pick the Right Tool 🔽

Classify each series and decide what to use.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.