Below are ten practice questions, each similar in style to an IB Math AA SL problem on sequences and series. They focus on arithmetic or geometric sequences. Step-by-step solutions are provided, with relevant formulas clearly stated.

Question 1 [6 marks]

An arithmetic sequence has first three terms \(u_1 = 20\), \(u_2 = 17\), and \(u_3 = 14\).

  1. [2 marks] Find the common difference \(d\).
  2. [2 marks] Find \(u_6\).
  3. [2 marks] Find the sum of the first 6 terms, \(S_6\).

Solution:

  1. Common difference \(d\)
    From an arithmetic sequence, \(d = u_2 - u_1 = 17 - 20 = -3\).
  2. Find \(u_6\)
    General term of an arithmetic sequence: \[ u_n = u_1 + (n-1)d. \] For \(n=6\): \[ u_6 = 20 + (6-1)(-3) = 20 + 5 \cdot (-3) = 20 - 15 = 5. \]
  3. Find \(S_6\)
    The sum of the first \(n\) terms in an arithmetic sequence is: \[ S_n = \frac{n}{2} \bigl(u_1 + u_n\bigr). \] So, \[ S_6 = \frac{6}{2}\bigl(u_1 + u_6\bigr) = 3 \times (20 + 5) = 3 \times 25 = 75. \]

Question 2 [6 marks]

An arithmetic sequence has first three terms \(u_1 = 7\), \(u_2 = 3\), and \(u_3 = -1\).

  1. [2 marks] Determine the common difference \(d\).
  2. [2 marks] Find \(u_7\).
  3. [2 marks] Calculate the sum of the first 7 terms, \(S_7\).

Solution:

  1. Common difference \(d\)
    Since \(u_2 - u_1 = 3 - 7 = -4\), we have \(d = -4\).
  2. Find \(u_7\)
    Using \(u_n = u_1 + (n-1)d\): \[ u_7 = 7 + (7-1)(-4) = 7 + 6 \cdot (-4) = 7 - 24 = -17. \]
  3. Find \(S_7\)
    Recall: \[ S_n = \frac{n}{2}(u_1 + u_n). \] Here, \(n=7\): \[ S_7 = \frac{7}{2} \bigl(7 + (-17)\bigr) = \frac{7}{2} \times (-10) = 7 \times (-5) = -35. \]

Question 3 [6 marks]

A geometric sequence has first three terms \(u_1 = 4\), \(u_2 = 12\), and \(u_3 = 36\).

  1. [2 marks] Find the common ratio \(r\).
  2. [2 marks] Find \(u_5\).
  3. [2 marks] Find the sum of the first 5 terms, \(S_5\).

Solution:

  1. Common ratio \(r\)
    In a geometric sequence, \(r = \frac{u_2}{u_1} = \frac{12}{4} = 3.\)
  2. Find \(u_5\)
    General term for a geometric sequence: \[ u_n = u_1 \, r^{\,n-1}. \] For \(n=5\): \[ u_5 = 4 \times 3^{\,5-1} = 4 \times 3^4 = 4 \times 81 = 324. \]
  3. Find \(S_5\)
    Sum of the first \(n\) terms in a geometric sequence (for \(r \neq 1\)): \[ S_n = u_1 \frac{r^n - 1}{r - 1}. \] Here \(n=5\), \(u_1=4\), and \(r=3\): \[ S_5 = 4 \times \frac{3^5 - 1}{3 - 1} = 4 \times \frac{243 - 1}{2} = 4 \times \frac{242}{2} = 4 \times 121 = 484. \]

Question 4 [6 marks]

A geometric sequence has \(u_1 = 16\), \(u_2 = 8\), and \(u_3 = 4\).

  1. [2 marks] Determine the common ratio \(r\).
  2. [2 marks] Find \(u_6\).
  3. [2 marks] Find the sum of the first 6 terms, \(S_6\).

Solution:

  1. Common ratio \(r\)
    Compute \(r = \frac{u_2}{u_1} = \frac{8}{16} = \tfrac{1}{2}.\)
  2. Find \(u_6\)
    \[ u_n = u_1 \, r^{\,n-1}. \] For \(n=6\): \[ u_6 = 16 \left(\tfrac{1}{2}\right)^{5} = 16 \times \tfrac{1}{32} = \tfrac{16}{32} = \tfrac{1}{2}. \]
  3. Find \(S_6\)
    Using \[ S_n = u_1 \frac{r^n - 1}{r - 1} \quad (\text{for }r \neq 1), \] we get \[ S_6 = 16 \times \frac{\left(\tfrac{1}{2}\right)^6 - 1}{\tfrac{1}{2} - 1} = 16 \times \frac{\tfrac{1}{64} - 1}{- \tfrac{1}{2}} = 16 \times \frac{- \tfrac{63}{64}}{- \tfrac{1}{2}}. \] Simplify step by step: \[ -\tfrac{63}{64} \div -\tfrac{1}{2} = \tfrac{63}{64} \times 2 = \tfrac{63}{32}. \] So \[ S_6 = 16 \times \tfrac{63}{32} = \tfrac{16 \times 63}{32} = \tfrac{16}{32} \times 63 = \tfrac{1}{2} \times 63 = 31.5. \] Or written as fraction: \[ S_6 = 31.5 \quad \text{or} \quad \tfrac{63}{2}. \]

Question 5 [6 marks]

Consider an arithmetic sequence where the first term \(u_1 = 5\) and the common difference \(d = 2\).

  1. [2 marks] Write down the first 3 terms of the sequence.
  2. [2 marks] Find \(u_{10}\), the 10th term.
  3. [2 marks] Calculate the sum of the first 10 terms, \(S_{10}\).

Solution:

  1. First 3 terms
    \(u_1 = 5.\)
    \(u_2 = 5 + 2 = 7.\)
    \(u_3 = 7 + 2 = 9.\)
  2. Find \(u_{10}\)
    General term: \(u_n = u_1 + (n-1)d.\)
    For \(n=10\): \[ u_{10} = 5 + (10-1)\times 2 = 5 + 9\times 2 = 5 + 18 = 23. \]
  3. Calculate \(S_{10}\)
    Sum formula: \[ S_n = \frac{n}{2} (u_1 + u_n). \] Here \(n=10\): \[ S_{10} = \frac{10}{2} (5 + 23) = 5 \times 28 = 140. \]

Question 6 [Maximum mark: 6]

Consider an arithmetic sequence: 5, 8, 11, 14, …

  1. (a) Find the common difference, d.
  2. (b) Find the 10th term in the sequence.
  3. (c) Find the sum of the first 10 terms in the sequence.

Solution:

(a) We see that \(u_1 = 5\) and \(u_2 = 8\). The common difference is
\(d = u_2 - u_1 = 8 - 5 = 3\).

(b) The nth term of an arithmetic sequence is \[ u_n = u_1 + (n - 1)d. \] For \(n = 10\): \[ u_{10} = 5 + (10 - 1)\times 3 = 5 + 9\times 3 = 5 + 27 = 32. \]

(c) The sum of the first n terms is \[ S_n = \frac{n}{2}\bigl(u_1 + u_n\bigr). \] Thus, for \(n = 10\): \[ S_{10} = \frac{10}{2} \bigl(5 + 32\bigr) = 5 \times 37 = 185. \]

Question 7 [Maximum mark: 6]

Consider an arithmetic sequence: 0, -4, -8, -12, …

  1. (a) Find the common difference, d.
  2. (b) Find the 10th term in the sequence.
  3. (c) Find the sum of the first 10 terms in the sequence.

Solution:

(a) We observe \(u_1 = 0\) and \(u_2 = -4\). Hence:
\(d = u_2 - u_1 = -4 - 0 = -4.\)

(b) Using \(\displaystyle u_n = u_1 + (n-1)d\):
\(u_{10} = 0 + (10-1)\times (-4) = 9 \times (-4) = -36.\)

(c) The sum of the first n terms formula: \(\displaystyle S_n = \frac{n}{2}\bigl(u_1 + u_n\bigr)\). For \(n=10\):
\(\displaystyle S_{10} = \frac{10}{2}\,\bigl(0 + (-36)\bigr) = 5 \times (-36) = -180.\)

Question 8 [Maximum mark: 6]

Consider an arithmetic sequence: 12, 9, 6, 3, …

  1. (a) Find the common difference, d.
  2. (b) Find the 10th term in the sequence.
  3. (c) Find the sum of the first 10 terms in the sequence.

Solution:

(a) Here \(u_1 = 12\) and \(u_2 = 9\). So
\(d = u_2 - u_1 = 9 - 12 = -3.\)

(b) Using \(u_n = u_1 + (n - 1)d\):
\(u_{10} = 12 + (10 - 1)\times (-3) = 12 + 9\times(-3) = 12 - 27 = -15.\)

(c) The sum formula: \(\displaystyle S_n = \frac{n}{2}(u_1 + u_n)\). So for \(n=10\):
\(\displaystyle S_{10} = \frac{10}{2}(12 + (-15)) = 5 \times (-3) = -15.\)

Question 9 [Maximum mark: 6]

Consider an arithmetic sequence: 1, 3, 5, 7, …

  1. (a) Find the common difference, d.
  2. (b) Find the 10th term in the sequence.
  3. (c) Find the sum of the first 10 terms in the sequence.

Solution:

(a) From \(u_1 = 1\) and \(u_2 = 3\), we get
\(d = 3 - 1 = 2.\)

(b) Using \(u_n = u_1 + (n-1)d\):
\(u_{10} = 1 + (10 - 1)\times 2 = 1 + 9\times 2 = 1 + 18 = 19.\)

(c) The sum of the first n terms: \(\displaystyle S_n = \frac{n}{2}(u_1 + u_n)\). Thus for \(n=10\):
\(\displaystyle S_{10} = \frac{10}{2}(1 + 19) = 5 \times 20 = 100.\)

Question 10 [Maximum mark: 6]

Consider an arithmetic sequence: 4, 10, 16, 22, …

  1. (a) Find the common difference, d.
  2. (b) Find the 10th term in the sequence.
  3. (c) Find the sum of the first 10 terms in the sequence.

Solution:

(a) Here \(u_1 = 4\) and \(u_2 = 10\). Hence
\(d = 10 - 4 = 6.\)

(b) Using the formula \(u_n = u_1 + (n-1)d\):
\(u_{10} = 4 + (10 - 1)\times 6 = 4 + 9\times 6 = 4 + 54 = 58.\)

(c) Recall the sum formula: \(\displaystyle S_n = \frac{n}{2}(u_1 + u_n)\). For \(n=10\):
\(\displaystyle S_{10} = \frac{10}{2}\,(4 + 58) = 5 \times 62 = 310.\)

Essential Formulas for Sequences & Series

Below are the essential formulas you need to master for IB Math AA SL questions on sequences and series. These formulas are frequently tested and form the foundation for solving practice problems.

Arithmetic Sequences

nth Term Formula: \(u_n = u_1 + (n-1)d\)

Where \(u_1\) = first term, \(d\) = common difference, \(n\) = term number

Common Difference: \(d = u_2 - u_1 = u_{n+1} - u_n\)

Sum of First n Terms:

\[S_n = \frac{n}{2}(u_1 + u_n) = \frac{n}{2}(2u_1 + (n-1)d)\]

Finding Term Number: Given \(u_n\), solve \(n = \frac{u_n - u_1}{d} + 1\)

Geometric Sequences

nth Term Formula: \(u_n = u_1 \cdot r^{n-1}\)

Where \(u_1\) = first term, \(r\) = common ratio, \(n\) = term number

Common Ratio: \(r = \frac{u_2}{u_1} = \frac{u_{n+1}}{u_n}\)

Sum of First n Terms: For \(r \neq 1\):

\[S_n = u_1 \cdot \frac{r^n - 1}{r - 1} = u_1 \cdot \frac{1 - r^n}{1 - r}\]

Sum to Infinity: For \(|r| < 1\):

\[S_\infty = \frac{u_1}{1 - r}\]

Special Series Formulas

Sum of First n Natural Numbers:

\[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\]

Sum of Squares:

\[1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\]

Sum of Cubes:

\[1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\]

Sigma Notation

Definition: \(\displaystyle \sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n\)

Properties:

  • \(\displaystyle \sum_{k=1}^{n} c \cdot a_k = c \cdot \sum_{k=1}^{n} a_k\) (Constant multiple)
  • \(\displaystyle \sum_{k=1}^{n} (a_k + b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k\) (Sum rule)
  • \(\displaystyle \sum_{k=1}^{n} c = nc\) (Constant sum)

Applications

Compound Interest: \(A = P(1 + r)^n\) (Geometric sequence application)

Linear Growth/Decay: \(A = P + nd\) (Arithmetic sequence application)

Loan Repayments: Combines geometric series with regular payments

Depreciation: \(V = V_0(1 - r)^n\) where \(r\) = depreciation rate

Frequently Asked Questions (FAQ)

1. What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference (d) to the previous term. For example, 3, 7, 11, 15, ... has a common difference of 4. The general formula is \(u_n = u_1 + (n-1)d\).
2. What is a geometric sequence?
A geometric sequence is a sequence where each term after the first is obtained by multiplying by a constant value called the common ratio (r). For example, 2, 6, 18, 54, ... has a common ratio of 3. The general formula is \(u_n = u_1 \times r^{n-1}\).
3. How do I find the common difference in an arithmetic sequence?
To find the common difference \(d\), subtract any term from the term that follows it: \(d = u_2 - u_1 = u_3 - u_2\), etc. For example, in the sequence 5, 8, 11, 14, ... the common difference is \(d = 8 - 5 = 3\).
4. How do I find the common ratio in a geometric sequence?
To find the common ratio \(r\), divide any term by the previous term: \(r = \frac{u_2}{u_1} = \frac{u_3}{u_2}\), etc. For example, in the sequence 4, 12, 36, 108, ... the common ratio is \(r = \frac{12}{4} = 3\).
5. What is the difference between a sequence and a series?
A sequence is an ordered list of numbers following a specific pattern (e.g., 2, 4, 6, 8, 10). A series is the sum of the terms in a sequence (e.g., 2 + 4 + 6 + 8 + 10 = 30). In IB notation, \(u_n\) refers to individual terms, while \(S_n\) refers to the sum.
6. What formulas should I memorize for IB Math AA SL?
Arithmetic: \(u_n = u_1 + (n-1)d\), \(S_n = \frac{n}{2}(u_1 + u_n)\)
Geometric: \(u_n = u_1 \times r^{n-1}\), \(S_n = u_1 \times \frac{r^n - 1}{r - 1}\), \(S_\infty = \frac{u_1}{1-r}\) for \(|r| < 1\)
These formulas are given in the IB formula booklet, but knowing them by heart speeds up problem-solving.
7. When does an infinite geometric series converge?
An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1, i.e., \(|r| < 1\). When this condition is met, the sum to infinity is \(S_\infty=\frac{u_1}{1-r}\). If \(|r| \geq 1\), the series diverges (sum is infinite).
8. How do I find which term equals a given value?
Set the nth term formula equal to the given value and solve for \(n\).
Arithmetic: Solve \(u_1 + (n-1)d = \text{value}\) for \(n\).
Geometric: Solve \(u_1 \times r^{n-1} = \text{value}\), then use logarithms: \(n = \frac{\log(\text{value}/u_1)}{\log(r)} + 1\).
9. What are common mistakes in sequence and series questions?
Common mistakes include: (1) Confusing \(u_n\) (nth term) with \(S_n\) (sum of n terms), (2) Using the wrong formula (arithmetic vs. geometric), (3) Calculator errors with exponents, (4) Forgetting that \(n\) must be a positive integer, (5) Not checking if \(|r| < 1\) before using \(S_\infty\).
10. How many marks are sequence questions worth in IB exams?
Sequence and series questions typically range from 6-10 marks in IB Math AA SL. They often appear as multi-part questions testing: (a) finding the common difference/ratio, (b) finding a specific term, (c) finding the sum of terms or sum to infinity, and sometimes (d) applications.
11. What is sigma notation?
Sigma notation \((\Sigma)\) is a compact way to write sums. For example, \(\sum_{k=1}^{5} k = 1 + 2 + 3 + 4 + 5 = 15\). The variable below \(\Sigma\) is the index, the number below is the starting value, and the number above is the ending value. The expression after \(\Sigma\) is what gets summed.
12. How do I know if a sequence is arithmetic or geometric?
Check the pattern: If the difference between consecutive terms is constant (e.g., +3, +3, +3), it's arithmetic. If the ratio of consecutive terms is constant (e.g., ×2, ×2, ×2), it's geometric. Some sequences are neither (e.g., Fibonacci, quadratic sequences).
13. What are real-world applications of sequences and series?
Arithmetic: Regular salary increases, linear depreciation, saving fixed amounts monthly.
Geometric: Compound interest, population growth, radioactive decay, loan repayments, bouncing ball heights.
IB exam questions often include these applications as word problems.
14. How do I find the sum when I don't know the last term?
For arithmetic series, use the alternative sum formula: \(S_n = \frac{n}{2}(2u_1 + (n-1)d)\). This only requires \(u_1\), \(d\), and \(n\). For geometric series, use \(S_n = u_1 \times \frac{r^n - 1}{r - 1}\), which also doesn't require knowing \(u_n\).
15. What is a recursive formula?
A recursive formula defines each term based on the previous term(s).
Arithmetic: \(u_n = u_{n-1} + d\) with \(u_1\) given.
Geometric: \(u_n = r \times u_{n-1}\) with \(u_1\) given.
This is different from the explicit formula which gives \(u_n\) directly.