Sequences and series

Think of it like a playlist of numbers! A sequence is just an ordered list of numbers, like songs in a playlist. Each number in the sequence is called a term (like a song).

For example, 2, 4, 6, 8, 10... is a sequence. Can you guess the next number? (It's 12!)

Now, what if you added up all the songs in your playlist? That's what a series is! A series is the sum of the terms in a sequence. So, for our example, 2 + 4 + 6 + 8 + 10 would be a series.

We'll mostly look at two main types of sequences and series: arithmetic (where you add the same number each time) and geometric (where you multiply by the same number each time). It's like having different types of music genres in your playlist!

Let's imagine you start a new savings plan. On the first day, you save $5. On the second day, you save $10. On the third day, $15, and so on. You always save $5 more than the day before.

Day 1: $5 Day 2: $10 Day 3: $15 Day 4: $20

This is an arithmetic sequence! Each day, you add $5 to the previous day's savings. The 'common difference' (the number you add each time) is $5.

Now, if you wanted to know how much money you've saved in total after 4 days, you'd add them up: $5 + $10 + $15 + $20 = $50. That's an arithmetic series! It's the sum of all the terms in your savings sequence.

An arithmetic sequence is like climbing a staircase where each step is the same height. You always add the same amount to get to the next number.

1. Identify the first term (a₁): This is where your sequence starts. Like the first step on the staircase.
2. Find the common difference (d): This is the fixed number you add (or subtract) to get from one term to the next. It's the height of each step.
3. Use the formula for the nth term: To find any term (like the 100th step without counting them all), we use: aₙ = a₁ + (n-1)d. Here, 'n' is the position of the term you want.
4. Example: If your sequence is 3, 7, 11, 15... a₁=3, d=4. The 5th term would be a₅ = 3 + (5-1)4 = 3 + 16 = 19.

A geometric sequence is like a chain reaction, where each step multiplies the previous one. Think of a rumour spreading: one person tells two, then those two tell two each, and so on.

1. Identify the first term (a₁): Again, this is your starting point.
2. Find the common ratio (r): This is the fixed number you multiply by to get from one term to the next. It's the 'spreading factor'.
3. Use the formula for the nth term: To find any term, we use: aₙ = a₁ * r^(n-1). Here, 'n' is the position of the term you want.
4. Example: If your sequence is 2, 6, 18, 54... a₁=2, r=3. The 4th term would be a₄ = 2 * 3^(4-1) = 2 * 3^3 = 2 * 27 = 54.

Let's break down how to find the sum of an arithmetic series (like your total savings).

1. Identify the type: First, figure out if it's arithmetic (adding) or geometric (multiplying).
2. List knowns: Write down a₁ (first term), d (common difference) or r (common ratio), and n (number of terms).
3. Choose the right formula: For arithmetic series, the sum (Sₙ) is Sₙ = n/2 * (2a₁ + (n-1)d). Or, if you know the last term (aₙ), Sₙ = n/2 * (a₁ + aₙ).
4. Substitute and calculate: Plug in your numbers and do the math carefully. It's like following a recipe!
5. Example: Sum of the first 4 terms of 5, 10, 15, 20... (a₁=5, d=5, n=4). S₄ = 4/2 * (25 + (4-1)5) = 2 * (10 + 35) = 2 * (10 + 15) = 2 * 25 = 50. Exactly what we found manually!

Here are some common traps students fall into and how to dodge them:

• Mixing up 'n' and 'n-1': Many formulas have (n-1). It's easy to forget the -1 or use 'n' instead. ❌ Using aₙ = a₁ + nd for arithmetic sequences. ✅ Remember it's (n-1) because the 'd' or 'r' is applied one less time than the number of terms.

• Calculator errors with exponents: When dealing with geometric sequences, r^(n-1) can be tricky, especially with negative 'r' values or large 'n'. ❌ Typing (-2)^4 as -2^4 (which is -(2^4) = -16). ✅ Use parentheses: (-2)^4 = 16. Always double-check your calculator input, especially with negative bases.

• Confusing sequences and series: Remember, a sequence is a list, a series is a sum. They are related but different! ❌ Answering a question asking for the 'sum' by just listing the terms. ✅ If it asks for the 'sum' or 'total', you need to add them up using the series formulas.

• Incorrectly identifying 'd' or 'r': Sometimes the pattern isn't immediately obvious. ❌ For 10, 5, 0, -5..., thinking d=5 instead of d=-5. ✅ Always check by subtracting (for 'd') or dividing (for 'r') a term by its previous term (e.g., a₂ - a₁ or a₂ / a₁).