Sequence, Series, Limit

Arithmetic

Sequence

Arithmetic math is based on addition. If you graph it, it always forms a perfectly straight line.

$a_n = a_1 + (n - 1)d$

Variable Breakdown:

• $a_n$ (The Target): The value of the number at position $n$. (e.g., “What is the 100th number?”)
• $a_1$ (The Start): The very first number in your list.
• $n$ (The Position): Which step you are on. We use $(n-1)$ because we don’t add the difference to the first term; we only start adding it from the second term onward.
• $d$ (The Common Difference): The amount you add (or subtract) at every step.

You start a job at $50,000 ($a_1$). You get a guaranteed raise of $3,000 ($d$) every year. What is your salary in Year 10 ($n$)?

$a_{10} = 50,000 + (10 - 1)3,000 = \mathbf{\$77,000}$

Series

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

Variable Breakdown:

• $S_n$: The “Sum.” The total of all numbers from the 1st to the $n$-th.
• $n/2$: This represents the number of “pairs” you have.
• $(a_1 + a_n)$: The sum of the first and last term.

Practical Example: A theater has 20 rows. The 1st row has 30 seats ($a_1$), and the 20th row has 80 seats ($a_{20}$). How many total seats ($S_{20}$) are in the theater?

$S_{20} = \frac{20(30 + 80)}{2} = 10 \times 110 = \mathbf{1,100 \text{ seats}}$

Geometric

Geometric math is based on multiplication. It starts slow but “explodes” or “vanishes” very quickly.

Sequence

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

Variable Breakdown:

• $a_1$: The starting value.
• $r$ (Common Ratio): What you multiply by each time.
  • If $r = 2$, it doubles.
  • If $r = 0.5$, it cuts in half.
• $n-1$: The number of times the growth has happened.

A viral post starts with 10 shares ($a_1$). The number of shares triples ($r=3$) every hour. How many shares will occur in Hour 6 ($n$)?

$a_6 = 10 \cdot 3^{(6-1)} = 10 \cdot 243 = \mathbf{2,430 \text{ shares}}$

Series

$S_n = a_1 \left( \frac{1 - r^n}{1 - r} \right)$

Variable Breakdown:

• $r^n$: The total growth factor over $n$ steps.
• $1-r$: The denominator that scales the sum correctly.

You save $100 this month. Every month you increase your savings by 10% ($r=1.10$). How much total have you saved after 4 months ($n$)?

$S_4 = 100 \left( \frac{1 - 1.10^4}{1 - 1.10} \right) = 100 \times 4.641 = \mathbf{\$464.10}$

Infinite Geometric Series

This is the “Limit” part of your heading. If a geometric sequence gets smaller and smaller ($|r| < 1$), the total sum doesn’t go to infinity—it hits a “wall” or a Limit.

$S_\infty = \frac{a_1}{1 - r}$

Variable Breakdown:

• $S_\infty$: The value the sum approaches but never exceeds.
• $1 - r$: The “gap” remaining.

You drop a “Super-Ball” from a height of 10 feet ($a_1$). Each bounce reaches half ($r=0.5$) the height of the previous bounce. If it bounces forever, what is the total vertical distance it travels?

$S_\infty = \frac{10}{1 - 0.5} = \frac{10}{0.5} = \mathbf{20 \text{ feet}}$

Limit

A limit describes the value that a function or sequence approaches as the input gets arbitrarily close to some point. Not every expression has a finite value at a given point, but it may approach one.

$\lim_{x \to a} f(x) = L$

• $L$: The value $f(x)$ gets closer to as $x$ gets closer to $a$.
• The arrow ($\to$) means “approaches”—$x$ gets arbitrarily close to $a$ without necessarily reaching it.

Connecting to Sequences

For sequences, the limit asks: as $n \to \infty$ (the term number goes to infinity), does $a_n$ approach a specific number?

• If $a_n$ settles toward a single value, the sequence converges to that limit.
• If $a_n$ grows without bound or oscillates forever, it diverges.

The infinite geometric series from the previous section is a classic example. As $n \to \infty$, $r^n \to 0$ (when $|r| < 1$), so the sum approaches the fixed value $\frac{a_1}{1 - r}$.