How to Solve: Sequences and Series Basics

How to Solve: Sequences and Series Basics

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Master sequences and series, and you’ll crack scholarship exams, predict stock trends, and even decode secret patterns in music—all while nailing those 5-mark questions on your algebra test."

What You Need To Know First

Before diving in, make sure you’re solid on: 1. Basic algebra (solving for n, substituting variables). 2. Exponents and roots (especially powers of 2, 3, and 10). 3. Linear equations (finding the nth term of a simple sequence like 2, 5, 8, 11…).

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

(MEMORISE THESE—most exams don’t provide them!)

1. Arithmetic Sequences

a) nth term formula: $ a_n = a_1 + (n - 1)d $ - $a_n$ = nth term - $a_1$ = first term - $d$ = common difference - $n$ = term number

b) Sum of first n terms: $ S_n = \frac{n}{2} (a_1 + a_n) $ OR $ S_n = \frac{n}{2} [2a_1 + (n - 1)d] $ - $S_n$ = sum of first n terms

2. Geometric Sequences

a) nth term formula: $ a_n = a_1 \cdot r^{n-1} $ - $r$ = common ratio

b) Sum of first n terms: $ S_n = a_1 \frac{1 - r^n}{1 - r} $ (if $r \neq 1$)

c) Sum of an infinite geometric series (only if $|r| < 1$): $ S_\infty = \frac{a_1}{1 - r} $

Step-by-Step Method

Follow these steps in order for any sequence/series problem.

Step 1: Identify the Type of Sequence

• Arithmetic? Check if the difference between terms is constant. Example: 2, 5, 8, 11… → $5 - 2 = 3$, $8 - 5 = 3$ → Arithmetic (d = 3).
• Geometric? Check if the ratio between terms is constant. Example: 3, 6, 12, 24… → $6/3 = 2$, $12/6 = 2$ → Geometric (r = 2).
• Neither? It might be a quadratic or other pattern (not covered here).

Step 2: Write Down What You Know

List: - First term ($a_1$) - Common difference (d) or ratio (r) - Term number (n) or number of terms to sum (n)

Step 3: Choose the Right Formula

• Need the nth term? → Use $a_n$ formula.
• Need the sum of terms? → Use $S_n$ formula.
• Infinite series? → Only use $S_\infty$ if $|r| < 1$.

Step 4: Plug in the Numbers and Solve

• Substitute known values into the formula.
• Solve for the unknown (e.g., n, $a_n$, $S_n$).

Step 5: Check Your Answer

• Does the answer make sense? (e.g., sum of positive terms can’t be negative).
• For nth term: Plug n back into the formula to verify.

Worked Example Using the Steps

Problem: Find the 10th term and the sum of the first 10 terms of the sequence: 4, 7, 10, 13…

Step 1: Identify the Type - Differences: $7 - 4 = 3$, $10 - 7 = 3$ → Arithmetic (d = 3).

Step 2: Write Down What You Know - $a_1 = 4$ - $d = 3$ - $n = 10$

Step 3: Choose the Right Formula - 10th term → $a_n = a_1 + (n - 1)d$ - Sum of first 10 terms → $S_n = \frac{n}{2} (a_1 + a_n)$

Step 4: Plug in the Numbers - 10th term: $a_{10} = 4 + (10 - 1) \cdot 3 = 4 + 27 = 31$ - Sum of first 10 terms: $S_{10} = \frac{10}{2} (4 + 31) = 5 \cdot 35 = 175$

Step 5: Check Your Answer - 10th term: $4 + 9 \cdot 3 = 31$ ✔️ - Sum: $4 + 7 + 10 + … + 31 = 175$ (count terms to confirm) ✔️

Final Answer: - 10th term = 31 - Sum of first 10 terms = 175

Worked Examples

Example 1 - Basic: Arithmetic Sequence

Problem: The 3rd term of an arithmetic sequence is 14, and the 7th term is 30. Find the first term and the common difference.

Step 1: Identify the Type - Given terms → Arithmetic (since it’s not geometric).

Step 2: Write Down What You Know - $a_3 = 14$ - $a_7 = 30$ - Need: $a_1$ and d.

Step 3: Use the nth Term Formula - For $a_3$: $a_1 + 2d = 14$ (1) - For $a_7$: $a_1 + 6d = 30$ (2)

Step 4: Solve the System of Equations - Subtract (1) from (2): $(a_1 + 6d) - (a_1 + 2d) = 30 - 14$ $4d = 16$ → $d = 4$ - Plug d into (1): $a_1 + 2(4) = 14$ → $a_1 = 6$

Step 5: Check - $a_3 = 6 + 2(4) = 14$ ✔️ - $a_7 = 6 + 6(4) = 30$ ✔️

Final Answer: - First term ($a_1$) = 6 - Common difference (d) = 4

What we did and why: We used the nth term formula for two terms to create equations, then solved for d first (easier) before finding $a_1$.

Example 2 - Medium: Geometric Sequence

Problem: A geometric sequence has a first term of 5 and a common ratio of 2. Find the sum of the first 6 terms.

Step 1: Identify the Type - Given r → Geometric.

Step 2: Write Down What You Know - $a_1 = 5$ - $r = 2$ - $n = 6$

Step 3: Choose the Right Formula - Sum of first n terms → $S_n = a_1 \frac{1 - r^n}{1 - r}$

Step 4: Plug in the Numbers $S_6 = 5 \cdot \frac{1 - 2^6}{1 - 2} = 5 \cdot \frac{1 - 64}{-1} = 5 \cdot 63 = 315$

Step 5: Check - Terms: 5, 10, 20, 40, 80, 160 - Sum: $5 + 10 + 20 + 40 + 80 + 160 = 315$ ✔️

Final Answer: Sum of first 6 terms = 315

What we did and why: We used the geometric sum formula because r ≠ 1. The denominator $(1 - r)$ cancels out the negative, making the calculation easier.

Example 3 - Exam Style: Disguised Problem

Problem: A savings account earns 5% interest compounded annually. If you deposit $1000 initially and add $1000 at the start of each year, how much will you have after 4 years? (Assume no withdrawals.)

Step 1: Identify the Pattern - Each year’s deposit grows by 5% → Geometric sequence for each deposit. - Total = Sum of 4 geometric sequences (one for each deposit).

Step 2: Break It Down - Deposit 1 (Year 0): $1000 grows for 4 years → $1000 \cdot (1.05)^4$ - Deposit 2 (Year 1): $1000 grows for 3 years → $1000 \cdot (1.05)^3$ - Deposit 3 (Year 2): $1000 grows for 2 years → $1000 \cdot (1.05)^2$ - Deposit 4 (Year 3): $1000 grows for 1 year → $1000 \cdot (1.05)^1$

Step 3: Recognize the Series - This is a geometric series with: - $a_1 = 1000 \cdot (1.05)^1$ (first term after 1 year) - $r = 1.05$ - $n = 4$ terms

Step 4: Use the Sum Formula $S_4 = 1000 \cdot (1.05)^1 \cdot \frac{1 - (1.05)^4}{1 - 1.05}$ $= 1000 \cdot 1.05 \cdot \frac{1 - 1.21550625}{-0.05}$ $= 1050 \cdot \frac{-0.21550625}{-0.05}$ $= 1050 \cdot 4.310125 = 4525.63$ (rounded to 2 decimal places)

Step 5: Add the Initial Deposit’s Growth - Initial $1000 grows for 4 years: $1000 \cdot (1.05)^4 = 1215.51$ - Total = 4525.63 + 1215.51 = 5741.14

Final Answer: Total after 4 years = $5741.14

What we did and why: We treated each deposit as a separate geometric sequence, summed them, and added the initial deposit’s growth. This is a real-world application of series!

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, listen up—this is your 60-second cheat sheet for sequences and series. First, spot the pattern: arithmetic sequences add the same number each time (d), geometric sequences multiply by the same number (r). For arithmetic, the nth term is $a_1 + (n-1)d$, and the sum is $\frac{n}{2}(a_1 + a_n)$. For geometric, the nth term is $a_1 \cdot r^{n-1}$, and the sum is $a_1 \frac{1 - r^n}{1 - r}$. Infinite sums? Only if $|r| < 1$, and the formula is $\frac{a_1}{1 - r}$. Watch out for off-by-one errors, negative d or r, and hidden patterns in word problems. Now go crush that exam!